Nonlinear Bending Analysis of Micro/Nano Rectangular and Annular Sector Plates Using a Modified Higher-Order Shear Deformation Theory and the Modified Couple Stress Theory
Authors
Abstract
In this paper, nonlinear bending analysis of functionally graded rectangular and sectorial micro/nano plates is investigated using the modified couple stress theory. For this purpose, a higher-order shear deformation theory and von Kármán geometrically nonlinear theory are employed. The equilibrium equations and the boundary conditions for rectangular and annular sector plates are derived from the principle of minimum total potential energy and solved using the Semi-Analytical Polynomial Method (SAPM). One of the advantages of the implemented shear deformation theory is removing the defects of higher order shear deformation theory, and obtaining the response of the first and the third-order shear deformation theories at the same time. Afterwards, beside investigating the benefits of this theory compared with other ones, the results are verified with those by other researches. At the end, the effects of length scale parameter, boundary conditions, power law index, and geometrical dimensions are investigated
Keywords
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15. Wang, Y. G., Hui-Lin, W., and Zhou, C. L., “Nonlinear Bending of Size-dependent Circular Micro-plates Based on the Modified Couple Stress Theory”, Archive of Applied Mechanics, Vol. 84, No. 1, pp. 391-400, 2013.
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17. Khorasani, V. S., and Bayat, M., “Bending Analysis of FG Plates Using a General Third-order Plate Theory with Modified Couple Stress Effect and MLPG Method”, Engineering Analysis with Boundary Elements, Vol. 94, No. 1, pp. 159-171, 2018.
18. Dastjerdi, Sh., Abbasi, M., and Yazdanparast, L., “A New Modified Higher-order Shear Deformation Theory for Nonlinear Analysis of Macro and Nano-annular Sector Plates Using the Extended Kantorovich Method in Conjunction with SAPM”, Acta Mechanica, Vol. 228, No. 1, pp. 3381-3401, 2017.
19. Kadoli, R., Akhtar, K., and Ganesan, N., “Static Analysis of Functionally Graded Beams Using Higher-order Shear Deformation Theory”, Applied Mathematical Modeling, Vol. 32, No. 1, pp. 2509-2525, 2008.
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, No. 1, pp. 78-87, 2016.
21. Touratier, M., “An Efficient Standard Plate Theory”, International Journal of Engineering Science, Vol. 29, No. 8, pp. 901-916, 1991.
22. Zenkour, A. M., “Generalized Shear Deformation Theory for Bending Analysis of Functionally Graded Plates”, Applied Mathematical Modeling, Vol. 30, No. 1, pp. 67-84, 2006.
23. Jomehzadeh, E., Noori, H. R., and Saidi, A. R., “The Size-dependent Vibration Analysis of Micro-plates Based on a Modified Couple Stress Theory”, Physica E, Vol. 43, No. 1, pp. 877-883, 2011.
24. Alinaghizadeh, F., Shariati, M., and Fish, J., “Bending Analysis of Size-dependent Functionally Graded Annular Sector Micro-plates Based on the Modified Couple Stress Theory”, Applied Mathematical Modeling, Vol. 44, No. 1, pp. 540-556, 2017.
25. Golkarian, A.R., Jabbarzadeh, M., and Dastjerdi, S., “A Novel Method for Numerical Analysis of 3D Nonlinear Thermo-mechanical Bending of Annular and Circular Plates with Asymmetric Boundary Conditions Using SAPM”, Journal of Solid Mechanics, Vol. 11, No. 3, pp. 498-512, 2019.
26. Thai, H. T., Choi, D. H., “Size-dependent Functionally Graded Kirchhoff and Mindlin Plate Models Based on a Modified Couple Stress Theory”, Composite Structures, Vol. 95, No. 1, pp. 142-153, 2013
2. Ichikawa, K., “Functionally Graded Materials in the 21st Century, A Workshop on Trends and Forecasts”, Springer Science and Business Media Newyork, 2001.
3. Arash, B., and Wang, Q., “A Review on the Application of Non-local Elastic Modes in Modeling of Carbon Nanotubes and Graphenes”, Computational Materials Science, Vol. 51, No. 1, pp. 303-313, 2012.
4. Levinson, M., “An Accurate Simple Theory of the Statics and Dynamics of Elastic Plate”, Mechanics Research Communication, Vol. 7, No. 6, pp. 343-350, 1980.
5. Reddy, J. N., and Liu, C. F., “A Higher-order Shear Deformation Theory of Laminated Elastic shells”, International Journal of Engineering Science, Vol. 23, No. 3, pp. 319-330, 1985.
6. Soldatos, K. P., “A Transverse Shear Deformation Theory for Homogeneous Monoclinic Plates”, Acta Mechanica, Vol. 94, No. 1, pp. 195-220, 1992.
7. Reissner, E., “On Transverse Bending of Plates Including the Effect of Transverse Shear Deformation”, International Journal of Solids and Structures, Vol. 11, No. 1, pp. 569-573, 1975.
8. Kaczkowski, Z., “Plates–Static Calculations”, Arkady, Warszawa, 1980.
9. Yang, F., Chong, A. C. M., Lam, D. C. C., and Tong, P., “Couple Stress Based Strain Gradient Theory for Elasticity”, International Journal of Solid and Structures, Vol. 39, No. 1, pp. 2731-2743, 2002.
10. Toupin, R. A., “Elastic Materials with Couple Stresses”, Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, pp. 385-414, 1962.
11. Mindlin, R. D. and Tiersten, H. F., “Effects of Couple-stresses in Linear Elasticity, “Archive for Rational Mechanics and Analysis, Vol. 11, No. 1, pp. 415-448, 1962.
12. Mindlin, R. D., “Micro-structures in Linear Elasticity”, Archive for Rational Mechanics and Analysis, Vol. 16, No. 1, pp. 51-78, 1964.
13. Reddy, J. N., and Jinseok, K., “A Nonlinear Modified Couple Stress-based Third-order Theory of Functionally Graded Plates”, Composite Structures, Vol. 94, No. 1, pp. 1128-1143, 2012.
14. Simsek, M., Kocatürk, T., and Akbas, S. D., “Static Bending of a Functionally Graded Micro-scale Timoshenko Beam Based on the Modified Couple Stress Theory”, Composite Structures, Vol. 95, No. 1, pp. 740-747, 2013.
15. Wang, Y. G., Hui-Lin, W., and Zhou, C. L., “Nonlinear Bending of Size-dependent Circular Micro-plates Based on the Modified Couple Stress Theory”, Archive of Applied Mechanics, Vol. 84, No. 1, pp. 391-400, 2013.
16. Lei, J., He, Y., Zhang, B., Liu, D., Shen, L., and Guo, S., “A Size-dependent FG Micro-plate Model Incorporating Higher-order Shear and Normal Deformation Effects Based on a Modified Couple Stress Theory”, International Journal of Mechanical Sciences, Vol. 104, No. 1, pp. 8-23, 2015.
17. Khorasani, V. S., and Bayat, M., “Bending Analysis of FG Plates Using a General Third-order Plate Theory with Modified Couple Stress Effect and MLPG Method”, Engineering Analysis with Boundary Elements, Vol. 94, No. 1, pp. 159-171, 2018.
18. Dastjerdi, Sh., Abbasi, M., and Yazdanparast, L., “A New Modified Higher-order Shear Deformation Theory for Nonlinear Analysis of Macro and Nano-annular Sector Plates Using the Extended Kantorovich Method in Conjunction with SAPM”, Acta Mechanica, Vol. 228, No. 1, pp. 3381-3401, 2017.
19. Kadoli, R., Akhtar, K., and Ganesan, N., “Static Analysis of Functionally Graded Beams Using Higher-order Shear Deformation Theory”, Applied Mathematical Modeling, Vol. 32, No. 1, pp. 2509-2525, 2008.
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, No. 1, pp. 78-87, 2016.
21. Touratier, M., “An Efficient Standard Plate Theory”, International Journal of Engineering Science, Vol. 29, No. 8, pp. 901-916, 1991.
22. Zenkour, A. M., “Generalized Shear Deformation Theory for Bending Analysis of Functionally Graded Plates”, Applied Mathematical Modeling, Vol. 30, No. 1, pp. 67-84, 2006.
23. Jomehzadeh, E., Noori, H. R., and Saidi, A. R., “The Size-dependent Vibration Analysis of Micro-plates Based on a Modified Couple Stress Theory”, Physica E, Vol. 43, No. 1, pp. 877-883, 2011.
24. Alinaghizadeh, F., Shariati, M., and Fish, J., “Bending Analysis of Size-dependent Functionally Graded Annular Sector Micro-plates Based on the Modified Couple Stress Theory”, Applied Mathematical Modeling, Vol. 44, No. 1, pp. 540-556, 2017.
25. Golkarian, A.R., Jabbarzadeh, M., and Dastjerdi, S., “A Novel Method for Numerical Analysis of 3D Nonlinear Thermo-mechanical Bending of Annular and Circular Plates with Asymmetric Boundary Conditions Using SAPM”, Journal of Solid Mechanics, Vol. 11, No. 3, pp. 498-512, 2019.
26. Thai, H. T., Choi, D. H., “Size-dependent Functionally Graded Kirchhoff and Mindlin Plate Models Based on a Modified Couple Stress Theory”, Composite Structures, Vol. 95, No. 1, pp. 142-153, 2013
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