Analytical Buckling of FG Nanobeams on The Basis of A New One Variable First-Order Shear Deformation Beam Theory

Year 2018, Volume: 10 Issue: 1, 21 - 34, 28.05.2018

Mohammad Malikan , Shahriar Dastjerdi

https://doi.org/10.24107/ijeas.420838

Cited By: 6

Abstract

In this work, buckling
analysis of functionally graded (FG) nanobeams based on a new refined beam theory
has been analyzed. The beam is modeled as an elastic beam subjected to
unidirectional compressive loads. To achieve this aim, the new obtained beam
theory has only one variable which lead to one equation similar to Euler beam
theory and also is free of any shear correction factor. The equilibrium
equation has been formulated by the nonlocal theory of Eringen to predict
small-scale effects. The equation has been solved by Navier’s approach by which critical
buckling loads have been obtained for simple boundaries. Finally, to approve
the results of the new beam theory, various beam theories have been compared.

Keywords

Buckling analysis, FG nanobeams, A new refined beam theory, Nonlocal elasticity theory, Navier’s approach

References

• De Volder, M. F., Tawfick, S. H., Baughman, R. H., Hart, A. J., Carbon nanotubes: present and future commercial applications. Science, 339, 535-539, 2013.
• Yu, M.-F., Lourie, O., Dyer, M. J., Moloni, K., Kelly, T. F., Ruoff, R. S., Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes Under Tensile Load. Science, 287, 637–640, 2000.
• Pop, E., Mann, D., Wang, Q., Goodson, K., Dai, H., Thermal conductance of an individual single-wall carbon nanotube above room temperature. Nano Letters, 6, 96–100, 2005.
• Sinha, S., Barjami, S., Iannacchione, G., Schwab, A., Muench, G., Off-axis thermal properties of carbon nanotube films. Journal of Nanoparticle Research, 7, 651–657, 2005.
• Koziol, K. K., Janas, D., Brown, E., Hao, L., Thermal properties of continuously spun carbon nanotube fibres. Physica E: Low-dimensional Systems and Nanostructures, 88, 104–108, 2017.
• Mintmire, J. W., Dunlap, B. I., White, C. T., Are Fullerene Tubules Metallic?. Physical Review Letters, 68, 631–634, 1992.
• Lu, X., Chen, Z., Curved Pi-Conjugation, Aromaticity, and the Related Chemistry of Small Fullerenes (C60) and Single-Walled Carbon Nanotubes. Chemical Reviews, 105, 3643–3696, 2005.
• Hilder, T. A., Hill, J. M., Modeling the Loading and Unloading of Drugs into Nanotubes. Small, 5, 300–308, 2009.
• Pastorin, G., Crucial Functionalizations of Carbon Nanotubes for Improved Drug Delivery: A Valuable Option?. Pharmaceutical Research, 26, 746–769, 2009.
• Bhirde, A. A., Patel, V., Gavard, J., Zhang, G., Sousa, A. A., Masedunskas, A., Leapman, R. D., Weigert, R., Gutkind, J. S., Rusling, J. F., Targeted Killing of Cancer Cells in Vivo and in Vitro with EGF-Directed Carbon Nanotube-Based Drug Delivery. ACS Nano, 3, 307–316, 2009.
• Reddy, J. N., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, 288–307, 2007.
• Gürses, M., Civalek, Ö., Korkmaz, A. K., Ersoy, H., Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first‐order shear deformation theory. International journal for numerical methods in engineering, 79, 290-313, 2009.
• Malikan, M., Jabbarzadeh, M., Dastjerdi, Sh., Non-linear Static stability of bi-layer carbon nanosheets resting on an elastic matrix under various types of in-plane shearing loads in thermo-elasticity using nonlocal continuum. Microsystem Technologies, 23, 2973-2991, 2017.
• Malikan, M., Buckling analysis of a micro composite plate with nano coating based on the modified couple stress theory. Journal of Applied and Computational Mechanics, 4, 1–15, 2018.
• Malikan, M., Analytical predictions for the buckling of a nanoplate subjected to nonuniform compression based on the four-variable plate theory, Journal of Applied and Computational Mechanics, 3, 218–228, 2017.
• Yao, X., Han, Q., The thermal effect on axially compressed buckling of a double-walled carbon nanotube. European Journal of Mechanics A/Solids, 26, 298–312, 2007.
• Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V., Darabi, M.A., Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin′s strain gradient theory. Applied Mathematical Modelling, 40, 9872-9891, 2016.
• Dai, H. L., Ceballes, S., Abdelkefi, A., Hong, Y. Z., Wang, L., Exact modes for post-buckling characteristics of nonlocal nanobeams in a longitudinal magnetic field. Applied Mathematical Modelling, 55, 758-775, 2018.
• Wang, B. L., Hoffman, M., Yu, A. B., Buckling analysis of embedded nanotubes using gradient continuum theory. Mechanics of Materials, 45, 52–60, 2012.
• Ke, L. L., Xiang, Y., Yang, J., Kitipornchai, S., Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. Computational Materials Science, 47, 409–417, 2009.
• Ansari, R., Sahmani, S., Rouhi, H., Axial buckling analysis of single-walled carbon nanotubes in thermal environments via the Rayleigh–Ritz technique. Computational Materials Science, 50, 3050–3055, 2011.
• Ansari, R., Faghih Shojaei, M., Mohammadi, V., Gholami, R., Rouhi, H., Buckling and postbuckling of single-walled carbon nanotubes based on a nonlocal Timoshenko beam model. ZAMM - Journal of Applied Mathematics and Mechanics, 1-13, 2014. https://doi.org/10.1002/zamm.201300017
• Ansari, R., Arjangpay, A., Nanoscale vibration and buckling of single-walled carbon nanotubes using the meshless local Petrov–Galerkin method. Physica E, 63, 283–292, 2014.
• Shen, H.-Sh., He, X.-Q., Yang, D.-Q., Vibration of thermally postbuckled carbon nanotube-reinforced composite beams resting on elastic foundations. International Journal of Non-Linear Mechanics, 91, 69-75, 2017.
• Mehralian, F., Tadi Beni, Y., Karimi Zeverdejani, M., Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes. Physica B: Condensed Matter, 514, 61-69, 2017.
• Wang, Y.-Z., Wang, Y.-S., Ke, L.-L., Nonlinear vibration of carbon nanotube embedded in viscous elastic matrix under parametric excitation by nonlocal continuum theory. Physica E: Low-dimensional Systems and Nanostructures, 83, 195-200, 2016.
• Baltacıoglu, A. K., Akgöz, B., Civalek, Ö., Nonlinear static response of laminated composite plates by discrete singular convolution method. Composite Structures, 93, 153-161, 2010.
• Reddy, J. N., Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids, 59, 2382–2399, 2011.
• Reddy, J. N., Arbind, A., Bending relationships between the modified couple stress-based functionally graded Timoshenko beams and homogeneous Bernoulli–Euler beams. Annals of Solid and Structural Mechanics, 3, 15–26, 2012.
• Mercan, K., Civalek, Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix. Composite Structures, 143, 300-309, 2016.
• Akgöz, B., Civalek, Ö., A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation. Composite Structures, 176, 1028-1038, 2017.
• Mercan, K., Numanoglu, H. M., Akgöz, B., Demir, C., Civalek, Ö., Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix. Archive of Applied Mechanics, 87, 1797-1814, 2017.
• Akgöz, B., Civalek, Ö., Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams. Composites Part B: Engineering, 129, 77-87, 2017.
• Dastjerdi, Sh., Lotfi, M., Jabbarzadeh, M., The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory. Composites Part B, 98, 78-87, 2016.
• Dastjerdi, Sh., Jabbarzadeh, M., Non-linear bending analysis of multi-layer orthotropic annular/circular graphene sheets embedded in elastic matrix in thermal environment based on non-local elasticity theory. Applied Mathematical Modelling, 41, 83–101, 2017.
• Dastjerdi, Sh., Jabbarzadeh, M., Nonlinear bending analysis of bilayer orthotropic graphene sheets resting on Winkler–Pasternak elastic foundation based on non-local continuum mechanics. Composites Part B, 87, 161-175, 2016.
• Malikan, M., Electro-mechanical shear buckling of piezoelectric nanoplate using modified couple stress theory based on simplified first order shear deformation theory. Applied Mathematical Modelling, 48, 196–207, 2017.
• Shimpi, R. P., Refined Plate Theory and Its Variants. AIAA JOURNAL, 40, 137-146, 2002.
• Malikan, M., Temperature influences on shear stability a nanosize plate with piezoelectricity effect. Multidiscipline modeling in materials and structures, 14, 125-142, 2017.
• Malikan, M., Sadraee Far, M. N., (2018), Differential quadrature method for dynamic buckling of graphene sheet coupled by a viscoelastic medium using neperian frequency based on nonlocal elasticity theory. Journal of Applied and Computational Mechanics, DOI: 10.22055/JACM.2017.22661.1138
• Ansari, R., Faghih Shojaei, M., Shahabodini, A., Bazdid-Vahdati, M., Three-dimensional bending and vibration analysis of functionally graded nanoplates by a novel differential quadrature-based approach. Composite Structures, 131, 753-764, 2015.
• Dastjerdi, Sh., Akgöz, B., New static and dynamic analyses of macro and nano FGM plates using exact three-dimensional elasticity in thermal environment. Composite Structures, 192, 626-641, 2018.
• Salehipour, H., Nahvi, H., Shahidi, A. R., Mirdamadi, H. R., 3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory. Applied Mathematical Modelling, 47, 174-188, 2017.
• Malikan, M., Nguyen, V. B., Buckling analysis of piezo-magnetoelectric nanoplates in hygrothermal environment based on a novel one variable plate theory combining with higher-order nonlocal strain gradient theory. Physica E: Low-dimensional Systems and Nanostructures, 102, 8-28, 2018.
• Wang, C. M., Zhang, Y. Y., Ramesh, S. S., Kitipornchai, S., Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory. Journal of Physics D: Applied Physics, 39, 3904-3909, 2006.
• Pradhan, S. C., Reddy, G. K., Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM. Computational Materials Science, 50, 1052–1056, 2011.
• Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E, 41, 1651-1655, 2009.
• Simsek, M., Yurtcu, H. H., Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Composite Structures, 97, 378–386, 2013.