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Natural Frequency and Buckling of Orthotropic Nanoplates Resting on Two-Parameter Elastic Foundations with Various Boundary Conditions

Published online by Cambridge University Press:  12 August 2014

M. Sobhy
M. Sobhy*
Affiliation:
Department of Mathematics, Faculty of Science, Kafrelsheikh University Kafrelsheikh, Egypt
*
*msobhy@sci.kfs.edu.eg, msobhy2011@gmail.com
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Abstract

In this article, the analyses of the natural frequency and buckling of orthotopic nanoplates, such as single-layered graphene sheets, resting on Pasternak's elastic foundations with various boundary conditions are presented. New functions for midplane displacements are suggested to satisfy the different boundary conditions. These functions are examined by comparing their results with the results obtained by using the functions suggested by Reddy (Reddy JN. Mechanics of Composite Materials and Structures: Theory and Analysis. Boca Raton, FL: CRC Press; 1997). Moreover, these functions are very simple comparing with Reddy's functions, leading to ease of calculations. The equations of motion of the nonlocal model are derived using the sinusoidal shear deformation plate theory (SPT) in conjunction with the nonlocal elasticity theory. The present SPT are compared with other plate theories. Explicit solution for buckling loads and vibration are obtained for single-layered graphene sheets with isotropic and orthotropic properties; and under biaxial loads. The formulation and the method of the solution are firstly validated by executing the comparison studies for the isotropic nanoplates with the results being in literature. Then, the influences of nonlocal parameter and the other parameters on the buckling and vibration frequencies are investigated.

Keywords

Natural frequencyBucklingNonlocal continuum modelOrthotopic nanoplatesPasternak's foundationsBoundary conditions
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 5 , October 2014 , pp. 443 - 453
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

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References

1.Eringen, A. C., “Nonlocal Polar Elastic Continua,” International Journal of Engineering Science, 10, pp. 1–16 (1972).Google Scholar
2.Eringen, A. C., “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves,” Journal of Applied Physics, 54, pp. 4703–4710 (1983).CrossRefGoogle Scholar
3.Eringen, A. C., Nonlocal Continuum Field Theories, Springer, New York (2002).Google Scholar
4.Ansari, R., Sahmani, S. and Arash, B., “Nonlocal Plate Model for Free Vibrations of Single-Layered Graphene Sheets,” Physics Letters A, 375, pp. 53–62 (2010).CrossRefGoogle Scholar
5.Pradhan, S. C. and Murmu, T., “Small Scale Effect on the Buckling of Single-Layered Graphene Sheets Under Biaxial Compression Via Nonlocal Continuum Mechanics,” Computers Materials Science, 47, pp. 268–274 (2009).Google Scholar
6.Murmu, T. and Pradhan, S. C., “Small-Scale Effect on the Free In-Plane Vibration of Nanoplates by Nonlocal Continuum Model,” Physica E, 41, pp. 1628–1633 (2009).Google Scholar
7.Pradhan, S. C. and Phadikar, J. K., “Nonlocal Elasticity Theory for Vibration of Nanoplates,” Journal of Sound and Vibration, 325, pp. 206–223 (2009).CrossRefGoogle Scholar
8.Pradhan, S. C. and Phadikar, J. K., “Small Scale Effect on Vibration of Embedded Multilayered Gra-phene Sheets Based on Nonlocal Continuum Models,” Physics Letters A, 373, pp. 1062–1069 (2009).Google Scholar
9.Ansari, R., Rajabiehfard, R. and Arash, B., “Nonlocal Finite Element Model for Vibrations of Embedded Multi-Layered Graphene Sheets,” Computers Materials Science, 49, pp. 831–838 (2010).Google Scholar
10.Sobhy, M., “Thermomechanical Bending and Free Vibration of Single-Layered Graphene Sheets Embedded in an Elastic Medium,” Physica E, 56, pp. 400–409 (2014).Google Scholar
11.Sobhy, M., “Generalized Two-Variable Plate Theory for Multi-Layered Graphene Sheets with Arbitrary Boundary Conditions,” Acta Mechanica, DOI: 10.1007/s00707-014-1093-5 (2014).Google Scholar
12.Zenkour, A. M. and Sobhy, M., “Nonlocal Elasticity Theory for Thermal Buckling of Nanoplates Lying on Winkler-Pasternak Elastic Substrate Medium,” Physica E, 53, pp. 251–259 (2013).Google Scholar
13.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 Structures, 105, pp. 163–172 (2013).CrossRefGoogle Scholar
14.Murmu, T., Pradhan, S. C., “Buckling of Biaxially Compressed Orthotropic Plates at Small Scales,” Mechanics Research Communications, 36, pp. 933–938 (2009).CrossRefGoogle Scholar
15.Malekzadeh, P., Setoodeh, A. R. and Beni, A. A., “Small Scale Effect on the Free Vibration of Ortho-tropic Arbitrary Straight-Sided Quadrilateral Nanoplates,” Composite Structures, 93, pp. 1631–1639 (2011).Google Scholar
16.Malekzadeh, P., Setoodeh, A. R. and Beni, A. A., “Small Scale Effect on the Thermal Buckling of Or-thotropic Arbitrary Straight-Sided Quadrilateral Na-noplates Embedded in an Elastic Medium,” Composite Structures, 93, pp. 2083–2089 (2011).Google Scholar
17.Khajeansari, A., Baradaran, G. H. and Yvonnet, J., “An Explicit Solution for Bending of Nanowires Lying on Winkler-Pasternak Elastic Substrate Medium Based on the Euler-Bernoulli Beam Theory,” International Journal of Engineering Science, 52, pp. 115–128 (2012).Google Scholar
18.Liew, K. M., He, X. Q. and Kitipornchai, S., “Predicting Nanovibration of Multi-Layered Graphene Sheets Embedded in an Elastic Matrix,” Acta Materialia, 54, pp. 4229–4236 (2006).Google Scholar
19.Murmu, T. and Pradhan, S. C., “Thermo-Mechanical Vibration of a Single-Walled Carbon Nanotube Embedded in an Elastic Medium Based on Nonlocal Elasticity Theory,” Computers Materials Science, 46, pp. 854–859 (2009).Google Scholar
20.Pradhan, S. C. and Murmu, T., “Small Scale Effect on the Buckling Analysis of Single-Layered Gra-phene Sheet Embedded in an Elastic Medium Based on Nonlocal Plate Theory,” Physica E, 42, pp. 1293–1301 (2010).CrossRefGoogle Scholar
21.Narendar, S. and Gopalakrishnan, S., “Critical Buckling Temperature of Single-Walled Carbon Nanotubes Embedded in a One-Parameter Elastic Medium Based on Nonlocal Continuum Mechanics,” Physica E, 43, pp. 1185–1191 (2011).CrossRefGoogle Scholar
22.Pradhan, S. C. and Reddy, G. K., “Buckling Analysis of Single Walled Carbon Nanotube on Winkler Foundation Using Nonlocal Elasticity Theory and DTM,” Computers Materials Science, 50, pp. 1052–1056 (2011).CrossRefGoogle Scholar
23.Samaei, A. T., Abbasion, S. and Mirsayar, M. M., “Buckling Analysis of a Single-Layer Graphene Sheet Embedded in an Elastic Medium Based on Nonlocal Mindlin Plate Theory,” Mechanics Research Communications, 38, pp. 481–485 (2011).Google Scholar
24.Mindlin, R. D., “Influence of Rotatory Inertia and Shear on Exural Motions of Isotropic Elastic Plates,” Journal of Applied Mechanics, 18, pp. 31–38 (1951).Google Scholar
25.Touratier, M., “An Efficient Standard Plate Theory,” International Journal of Engineering Science, 29, pp. 901–916 (1991).Google Scholar
26.Reddy, J. N., “Analysis of Functionally Graded Plates,” International Journal for Numerical Methods in Engineering, 47, pp. 663–684 (2000).Google Scholar
27.Aydogdu, M., “A General Nonlocal Beam Theory: Its Application to Nanobeam Bending, Buckling and Vibration,” Physica E, 41, pp. 1651–1655 (2009).Google Scholar
28.Aghababaei, R. and Reddy, J. N., “Nonlocal Third-Order Shear Deformation Plate Theory with Application to Bending and Vibration of Plates,” Journal of Sound and Vibration, 326, pp. 277–289 (2009).Google Scholar
29.Thai, H. T., “A nonlocal Beam Theory for Bending, Buckling, and Vibration of Nanobeams,” International Journal of Engineering Science, 52, pp. 56–64 (2012).Google Scholar
30.Zenkour, A. M. and Sobhy, M., “Thermal Buckling of Various Types of FGM Sandwich Plates,” Composite Structures, 93, pp. 93–102 (2010).Google Scholar
31.Zenkour, A. M. and Sobhy, M., “Thermal Buckling of Functionally Graded Plates Resting on Elastic Foundations Using the Trigonometric Theory,” Journal of Thermal Stresses, 34, pp. 1119–1138 (2011).Google Scholar
32.Zenkour, A. M. and Sobhy, M., “Elastic Foundation Analysis of Uniformly Loaded Functionally Graded Viscoelastic Sandwich Plates,” Journal of Mechanical, 28, pp. 439–452 (2012).Google Scholar
33.Sobhy, M., “Buckling and Free Vibration of Exponentially Graded Sandwich Plates Resting on Elastic Foundations Under Various Boundary Conditions,” Composite Structures, 99, pp. 76–87 (2013).CrossRefGoogle Scholar
34.Reddy, J. N., Mechanics of Composite Materials and Structures: Theory and Analysis, FL: CRC Press, Boca Raton (1997).Google Scholar
35.Reddy, J. N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed., FL: CRC Press, Boca Raton (2004).Google Scholar