The Nonlinear Bending Analysis for Circular Nano Plates Based on Modified Coupled Stress and Three- Dimensional Elasticity Theories
Authors
Abstract
In this paper, the nonlinear bending analysis for annular circular nano plates is conducted based on the modified coupled stress and three-dimensional elasticity theories. For this purpose, the equilibrium equations, considering nonlinear strain terms, are calculated using the least energy potential method and solved by the numerical semi-analytical polynomial method. According to the previous works, there have been no studies calculating all boundary conditions numerically based on three-dimensional elasticity. Typically, the research done on three-dimensional elasticity is either finite element or only for a simply-supported boundary condition. In this research, for the first time, the nonlinear analysis of bending is calculated with the help of three-dimensional elasticity for a variety of boundary conditions. Also, with the help of the modified couple stress theory, the results on the nano-scale scale have been studied. In the following, while validating the results, we investigate the changes in the scale parameter for the types of boundary conditions, the effect of changing the parameter of scale in different thicknesses, and the impact of the parameter of scale on the linear and nonlinear results.
Keywords
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13. Salehipour, H., Nahvi, H., and Shahidi, A. R., “Exact Closed-Form Free Vibration Analysis for Functionally Graded Micro/Nano Plates Based on Modified Couple Stress and Three-Dimensional Elasticity Theories”, Composite Structures, Vol. 124, pp. 283-291, 2015.
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17. Atashipour, S. R., Girhammar, U. A., and Al-Emrani, M., “Exact Lévy-type Solutions for Bending of Thick Laminated Orthotropic Plates Based on 3-D Elasticity and Shear Deformation Theories”, Composite Structures, Vol. 163, pp. 129-151, 2017.
18. Shaban, M., and Alibeigloo, A., “Three-Dimensional Elasticity Solution for Sandwich Panels with Corrugated Cores by Using Energy Method”, Thin-Walled Structures, Vol. 119, pp. 404-411, 2017.
19. Asemi, K., Salehi, M., and Akhlaghi, M., “Post-Buckling Analysis of FGM Annular Sector Plates Based on Three Dimensional Elasticity Graded Finite Elements”, International Journal of Non-Linear Mechanics, Vol. 67, pp. 164-177, 2014.
20. Dastjerdi, S., Lotfi, M., and Jabbarzadeh, M., “The Effect of Vacant Defect on Bending Analysis of Graphene Sheets Based on the Mindlin Nonlocal Elasticity Theory”, Composites Part B, Vol. 98, pp. 78-87, 2016.
21. Dastjerdi, S., Jabbarzadeh, M., and Aliabadi, S., “Nonlinear Static Analysis of Single Layer Annular/Circular Graphene Sheets Embedded in Winkler–Pasternak Elastic Matrix Based on Non-Local Theory of Eringen”, Ain Shams Engineering Journal, Vol. 7, pp. 873-884, 2016.
22. Wang, Y. G., Lin, W. H., and Zhou, C. L., “Nonlinear Bending of Size-Dependent Circular Micro Plates Based on the Modified Couple Stress Theory”, Applied Mechanics, Vol. 84, pp. 391-400, 2014.
2. Eringen, A. C., Nonlocal Continuum Field Theories, Springer-Verlag, NewYork, Inc., pp. 71-175, 2002.
3. Liang, X., Hu, S., and Shen, S., “A New Bernoulli–Euler Beam Model Based on a Simplified Strain Gradient Elasticity Theory and its Applications”, Composite Structures, Vol. 111, No. 1, pp. 317-323, 2014.
4. Shakouri, A., Ng, T. Y., and Lin, R. M., “A Study of the Scale Effects on the Flexural Vibration of Graphene Sheets Using REBO Potential Based Atomistic Structural and Nonlocal Couple Stress Thin Plate Models”, Physica E: Low-dimensional Systems and Nanostructures, Vol. 50, No. 1, pp. 22-28, 2013.
5. Ashoori, A., and Mahmoodi, M. J., “The Modified Version of Strain Gradient and Couple Stress Theories in General Curvilinear Coordinates”, European Journal of Mechanics - A/Solids, Vol. 49, No. 1, pp. 441-454, 2015.
6. Eringen, A. C., and Edelen, D. G. B., “On Nonlocal Elasticity”, International Journal of Engineering Science, Vol. 10, No. 1, pp. 233-248, 1972.
7. Golmakani, M. E., and Rezatalab, J., “Nonlinear Bending Analysis of Orthotropic Nanoplates Based on Nonlocal Model of Eringen Using DQM”, Modares Mechanical Engineering, Vol. 13, No. 12, pp. 122-136, 2012 (in Persian).
8. Jabbarzadeh, M., Talati, H., and Noroozi, A. R., “Nonlinear Analysis of Circular Graphene Sheet Using Nonlocal Continuum Mechanic Theory”, Modares Mechanical Engineering, Vol. 13, No. 13, pp. 57-66, 2012 (in Persian).
9. Alibeigloo, A., and Liew, K. M., “Free Vibration Analysis of Sandwich Cylindrical Panel with Functionally Graded Core Using Three-Dimensional Theory of Elasticity”, Composite Structures, Vol. 113, pp. 23-30, 2014.
10. Salehipour, H., Nahvi, H., and Shahidi, A. R., “Closed-Form Elasticity Solution for Three-Dimensional Deformation of Functionally Graded Micro/ Nano Plates on Elastic Foundation”, Latin American Journal of Solids and Structures, Vol. 12, No. 4, 2015.
11. Salehipour, H., Nahvi, H., Shahidi, A. R., and 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, Vol. 47, pp. 174-188, 2017.
12. Salehipour, H., Nahvi, H., and Shahidi, A. R., “Exact Analytical Solution for Free Vibration of Functionally Graded Micro/Nanoplates Via Three-Dimensional Nonlocal Elasticity”, Physica E, Vol. 66, pp. 350-358, 2015.
13. Salehipour, H., Nahvi, H., and Shahidi, A. R., “Exact Closed-Form Free Vibration Analysis for Functionally Graded Micro/Nano Plates Based on Modified Couple Stress and Three-Dimensional Elasticity Theories”, Composite Structures, Vol. 124, pp. 283-291, 2015.
14. Zafarmand, H., and Kadkhodayan, M., “Three Dimensional Elasticity Solution for Static and Dynamic Analysis of Multi-Directional Functionally Graded Thick Sector Plates with General Boundary Conditions”, Composites: Part B, Vol. 69, pp. 592-602, 2015.
15. Ansari, R., Shahabodini, A., and Faghih Shojaei, M., “Nonlocal Three-Dimensional Theory of Elasticity with Application to Free Vibration of Functionally Graded Nanoplates on Elastic Foundations”, Physica E, Vol. 76, pp. 70-81, 2016.
16. Adineh, M., and Kadkhodayan, M., “Three-Dimensional Thermo-Elastic Analysis and Dynamic Response of a Multi-Directional Functionally Graded Skew Plate on Elastic Foundation”, Composites Part B, Vol. 125, pp. 227-240, 2017.
17. Atashipour, S. R., Girhammar, U. A., and Al-Emrani, M., “Exact Lévy-type Solutions for Bending of Thick Laminated Orthotropic Plates Based on 3-D Elasticity and Shear Deformation Theories”, Composite Structures, Vol. 163, pp. 129-151, 2017.
18. Shaban, M., and Alibeigloo, A., “Three-Dimensional Elasticity Solution for Sandwich Panels with Corrugated Cores by Using Energy Method”, Thin-Walled Structures, Vol. 119, pp. 404-411, 2017.
19. Asemi, K., Salehi, M., and Akhlaghi, M., “Post-Buckling Analysis of FGM Annular Sector Plates Based on Three Dimensional Elasticity Graded Finite Elements”, International Journal of Non-Linear Mechanics, Vol. 67, pp. 164-177, 2014.
20. Dastjerdi, S., Lotfi, M., and Jabbarzadeh, M., “The Effect of Vacant Defect on Bending Analysis of Graphene Sheets Based on the Mindlin Nonlocal Elasticity Theory”, Composites Part B, Vol. 98, pp. 78-87, 2016.
21. Dastjerdi, S., Jabbarzadeh, M., and Aliabadi, S., “Nonlinear Static Analysis of Single Layer Annular/Circular Graphene Sheets Embedded in Winkler–Pasternak Elastic Matrix Based on Non-Local Theory of Eringen”, Ain Shams Engineering Journal, Vol. 7, pp. 873-884, 2016.
22. Wang, Y. G., Lin, W. H., and Zhou, C. L., “Nonlinear Bending of Size-Dependent Circular Micro Plates Based on the Modified Couple Stress Theory”, Applied Mechanics, Vol. 84, pp. 391-400, 2014.
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