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Investigation of Vibration and Thermal Buckling of Nanobeams Embedded in An Elastic Medium Under Various Boundary Conditions

Published online by Cambridge University Press:  18 November 2015

D. S. Mashat
Affiliation:
Department of MathematicsFaculty of ScienceKing Abdulaziz UniversityJeddah, Saudi Arabia
A. M. Zenkour
Affiliation:
Department of MathematicsFaculty of ScienceKing Abdulaziz UniversityJeddah, Saudi Arabia Department of MathematicsFaculty of ScienceKafrelsheikh UniversityKafrelsheikh, Egypt
M. Sobhy*
Affiliation:
Department of MathematicsFaculty of ScienceKafrelsheikh UniversityKafrelsheikh, Egypt Department of Mathematics and StatisticsFaculty of ScienceKing Faisal UniversityHofuf, Saudi Arabia
*
*Corresponding author ([email protected], [email protected])
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Abstract

Analyses of free vibration and thermal buckling of nanobeams using nonlocal shear deformation beam theories under various boundary conditions are precisely illustrated. The present beam is restricted by vertically distributed identical springs at the top and bottom surfaces of the beam. The equations of motion are derived using the dynamic version of Hamilton's principle. The governing equations are solved analytically when the edges of the beam are simply supported, clamped or free. Thermal buckling solution is formulated for two types of temperature change through the thickness of the beam: Uniform and linear temperature rise. To validate the accuracy of the results of the present analysis, the results are compared, as possible, with solutions found in the literature. Furthermore, the influences of nonlocal coefficient, stiffness of Winkler springs and span-to-thickness ratio on the frequencies and thermal buckling of the embedded nanobeams are examined.

Type
Research Article
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2016 

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References

1.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. 12441255 (2011).CrossRefGoogle Scholar
2.Esawi, A. M. K. and Farag, M. M., “Carbon Nanotube Reinforced Composites: Potential and Current Challenges,” Materials and Design, 28, pp. 23942401 (2007).CrossRefGoogle Scholar
3.Katsnelson, M. I. and Novoselov, K. S., “Graphene: New Bridge Between Condensed Matter Physics and Quantum Electrodynamics,” Solid State Communications, 143, pp. 313 (2007).CrossRefGoogle Scholar
4.Toupin, R. A., “Elastic Materials with Couple-Stresses,” Archive for Rational Mechanics and Analysis, 11, pp. 385414 (1962).Google Scholar
5.Eringen, A. C. and Suhubi, E. S., “Nonlinear Theory of Simple Micro-Elastic Solids-I,” International Journal of Engineering Science, 2, pp. 189203 (1964).CrossRefGoogle Scholar
6.Fleck, N. A. and Hutchinson, J. W., “Strain Gradient Plasticity,” Advances in Applied Mechanics, 33, pp. 295361 (1997).CrossRefGoogle Scholar
7.Eringen, A. C., “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves,” Journal of Applied Physics, 54, pp. 47034710 (1983).CrossRefGoogle Scholar
8.Eringen, A. C., Nonlocal Continuum Field Theories, Springer-Verlag, New York (2002).Google Scholar
9.Reddy, J. N., “Nonlocal Theories for Bending, Buckling and Vibration of Beams,” International Journal of Engineering Science, 45, pp. 288307 (2007).CrossRefGoogle Scholar
10.Roque, C. M. C., Ferreira, A. J. M. and Reddy, J. N., “Analysis of Timoshenko Nanobeams with a Nonlocal Formulation and Meshless Method,” International Journal of Engineering Science, 49, pp. 976984 (2011).Google Scholar
11.Thai, H. T., “A Nonlocal Beam Theory for Bending, Buckling, and Vibration of Nanobeams,” International Journal of Engineering Science, 52, pp. 5664 (2012).Google Scholar
12.Thai, H. T., Vo, T. P., Nguyen, T. K. and Lee, J., “A Nonlocal Sinusoidal Plate Model for Micro/Nanoscale Plates,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228, pp. 26522660 (2014).Google Scholar
13.Setoodeh, A. R., Malekzadeh, P. and Vosoughi, A. R., “Nonlinear Free Vibration of Orthotropic Graphene Sheets Using Nonlocal Mindlin Plate Theory,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 226, pp. 18961906 (2011).Google Scholar
14.Babaei, H. and Shahidi, A., “Vibration of Quadrilateral Embedded Multilayered Graphene Sheets Based on Nonlocal Continuum Models Using the Galerkin Method,” Acta Mechanica Sinica, 27, pp. 967976 (2011).Google Scholar
15.Haghshenas, A. and Arani, A. G., “Nonlocal Vibration of a Piezoelectric Polymeric Nanoplate Carrying Nanoparticle Via Mindlin Plate Theory,” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 226, pp. 18961906 (2014).Google Scholar
16.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. 163172 (2013).Google Scholar
17.Sobhy, M., “Thermomechanical Bending and Free Vibration of Single-Layered Graphene Sheets Embedded in an Elastic Medium,” Physica E, 56, pp. 400409 (2014).CrossRefGoogle Scholar
18.Sobhy, M., “Generalized Two-Variable Plate Theory for Multi-Layered Graphene Sheets with Arbitrary Boundary Conditions,” Acta Mechanica, 225, pp. 25212538 (2014).Google Scholar
19.Ke, L. L., Wang, Y. S., Yang, J. and Kitipornchai, S., “Free Vibration of Size-Dependent Magneto-Electro-Elastic Nanoplates Based on the Nonlocal Theory,” Acta Mechanica Sinica, 30, pp. 516525 (2014).Google Scholar
20.Levinson, M., “A New Rectangular Beam Theory,” Journal of Sound Vibration, 74, pp. 8187 (1981).Google Scholar
21.Touratier, M., “An Efficient Standard Plate Theory,” International Journal of Engineering Science, 29(8), pp. 901916 (1991).Google Scholar
22.Zenkour, A. M., “Analytical Solution for Bending of Cross-Ply Laminated Plates Under Thermo-Mechanical Loading,” Composite Structures, 65, pp. 367379 (2004).CrossRefGoogle Scholar
23.Zenkour, A. M., “A Comprehensive Analysis of Functionally Graded Sandwich Plates: Part 1-Deflection and Stresses and Part 2-Buckling and Free Vibration,” International Journal of Solids and Structures, 42, pp. 52245258 (2005).Google Scholar
24.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. 251259 (2013).Google Scholar
25.Reddy, J. N., “Analysis of Functionally Graded Plates,” International Journal for Numerical Methods in Engineering, 47, pp. 663684 (2000).3.0.CO;2-8>CrossRefGoogle Scholar
26.Phadikar, J. K. and Pradhan, S. C., “Variational Formulation and Finite Element Analysis for Nonlocal Elastic Nanobeams and Nanoplates,” Computational Materials Science, 49, pp. 492499 (2010).Google Scholar
27.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. 11851191 (2011).CrossRefGoogle Scholar
28.Ke, L. L., Xiang, Y., Yang, J. and Kitipornchai, S., “Nonlinear Free Vibration of Embedded Double-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory,” Computational Materials Science, 47, pp. 409417 (2009).CrossRefGoogle Scholar
29.Behfar, K. and Naghdabadi, R., “Nanoscale Vibrational Analysis of a Multi-Layered Graphene Sheet Embedded in an Elastic Medium,” Composites Science and Technology, 65, pp. 11591164 (2005).Google Scholar
30.Malekzadeh, P., Setoodeh, A. R. and Beni, A. A., “Small Scale Effect on the Thermal Buckling of Orthotropic Arbitrary Straight-Sided Quadrilateral Nanoplates Embedded in an Elastic Medium,” Composite Structures, 93, pp. 20832089 (2011).Google Scholar
31.Xu, F., Durham, J. W., Wiley, B. J. and Zhu, Y., “Strain-Release Assembly of Nanowires on Stretchable Substrates,” ACS Nano, 5, pp. 15561563 (2011).Google Scholar
32.Shan, W. L. and Chen, Z., “Mechanical Instability of Thin Elastic Rods,” Journal of Postdoctoral Research, 1, pp. 18 (2013).Google Scholar
33.Xiao, J. and Chen, X., “Buckling Morphology of an Elastic Beam Between Two Parallel Lateral Constraints: Implication for a Snake Crawling Between Walls,” Journal of Royal Society Interface, 10, pp. 16 (2013).CrossRefGoogle ScholarPubMed
34.Brangwynne, C. P., MacKintosh, F. C., Kumar, S., Geisse, N. A., Talbot, J., Mahadevan, L.et al., “Microtubules Can Bear Enhanced Compressive Loads in Living Cells Because of Lateral Reinforcement,” Journal of Cell Biology, 173, pp. 733741 (2006).CrossRefGoogle ScholarPubMed
35.Shan, W. L., Chen, Z., Broedersz, C. P., Gumaste, A. A., Soboyejo, W. O. and Brangwynne, C. P., “Attenuated Short Wavelength Buckling and Force Propagation in a Biopolymer-Reinforced Rod,” Soft Matter, 9, 194199 (2013).Google Scholar
36.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. 11191138 (2011).CrossRefGoogle Scholar
37.Aghababaei, R. and Reddy, J. N., “Nonlocal Third-Order Shear Deformation Plate Theory with Application to Bending and Vibration of Plates,” Journal of Sound Vibration, 326, pp. 277289 (2009).Google Scholar
38.Sobhy, M., “Buckling and Free Vibration of Exponentially Graded Sandwich Plates Resting on Elastic Foundations Under Various Boundary Conditions,” Composite Structures, 99, pp. 7687 (2013).CrossRefGoogle Scholar
39.Sobhy, M., “Natural Frequency and Buckling of Orthotropic Nanoplates Resting on Two-Parameter Elastic Foundations with Various Boundary Conditions,” Journal of Mechanics, 30, pp. 443453 (2014).Google Scholar
40.Reddy, J. N., Mechanics of Composite Materials and Structures: Theory and Analysis, CRC Press, Boca Raton (1997).Google Scholar
41.Roque, C. M. C., Ferreira, A. J. M. and Reddy, J. N., “Analysis of Timoshenko Nanobeams with a Nonlocal Formulation and Meshless Method,” International Journal of Engineering Science, 49, pp. 976984 (2011).Google Scholar