Large Deflection Analysis of Graphene Sheets in the Thermal Environment Using Higher-Order Nonlocal Strain Gradient Principle

Document Type : Research Article

Authors

Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract

This paper investigates the large deflection of a sector nanoplate on the Winkler elastic foundation in the thermal environment based on the nonlocal strain gradient principle. By taking into account von Karman’s nonlinear strains and applying the higher-order shear deformation theory (HSDT), the governing equations of the graphene plate are derived. By presenting acceptable accuracy without the need for a shear correction coefficient, HSDT eliminates the defects of the first shear deformation theory (FSDT) and provides an appropriate distribution for shear stress along the thickness. The equations have been solved using the differential quadrature method (DQM) and the extended Kantorovich method (EKM). The results of the present study are compared with the available references, which demonstrate good agreement among them. For example, the results of the present study for the radius ratios of 0.25, 0.5, and 0.75 have 0.35%, 2.83%, and 7% differences with Ref. [1]. In conclusion, this study examines the impact of various small-scale parameters, load, boundary conditions, geometric dimensions, and elastic foundation on the maximum nondimensional deflection in the thermal environment.

Keywords

Main Subjects

References

[1]   Harik, I.E., 1984. Analytical solution to the orthotropic sector. Journal of Engineering Mechanics, 110 (4), pp.554-568.
[2]   Kumar, R., Kumar, M. & Luthra, G., 2023. Fundamental approaches and applications of nanotechnology: A mini review. Materials Today: Proceedings.
[3]   Farajpour, A., Mohammadi, M., Shahidi, A.R. & Mahzoon, M., 2011. Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model. Physica E: Low-dimensional Systems and Nanostructures, 43 (10), pp.1820-1825.
[4]   Eringen, A.C., 1972. Nonlocal polar elastic continua. International Journal of Engineering Science, 10 (1), pp.1-16.
[5]   Trabelsi, S., Frikha, A., Zghal, S. & Dammak, F., 2018. Thermal post-buckling analysis of functionally graded material structures using a modified fsdt. International Journal of Mechanical Sciences, 144, pp.74-89.
[6]   Trabelsi, S., Frikha, A., Zghal, S. & Dammak, F., 2019. A modified fsdt-based four nodes finite shell element for thermal buckling analysis of functionally graded plates and cylindrical shells. Engineering Structures, 178, pp.444-459.
[7]   Zghal, S., Frikha, A. & Dammak, F., 2020. Large deflection response-based geometrical nonlinearity of nanocomposite structures reinforced with carbon nanotubes. Applied Mathematics and Mechanics, 41 (8), pp.1227-1250.
[8]   Joueid, N., Zghal, S., Chrigui, M. & Dammak, F., 2023. Thermoelastic buckling analysis of plates and shells of temperature and porosity dependent functionally graded materials. Mechanics of Time-Dependent Materials.
[9]   Zghal, S., Joueid, N., Tornabene, F., Dimitri, R., Chrigui, M. & Dammak, F., 2024. Time-dependent deflection responses of fg porous structures subjected to different external pulse loads. Journal of Vibration Engineering & Technologies, 12 (1), pp.857-876.
[10] Trabelsi, S., Zghal, S. & Dammak, F., 2020. Thermo-elastic buckling and post-buckling analysis of functionally graded thin plate and shell structures. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 42 (5), pp.233.
[11] Zghal, S., Frikha, A. & Dammak, F., 2018. Free vibration analysis of carbon nanotube-reinforced functionally graded composite shell structures. Applied Mathematical Modelling, 53, pp.132-155.
[12] Zghal, S. & Dammak, F., 2020. Vibrational behavior of beams made of functionally graded materials by using a mixed formulation. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 234 (18), pp.3650-3666.
[13] Zghal, S., Trabelsi, S. & Dammak, F., 2022. Post-buckling behavior of functionally graded and carbon-nanotubes based structures with different mechanical loadings. Mechanics Based Design of Structures and Machines, 50 (9), pp.2997-3039.
[14] Zghal, S., Frikha, A. & Dammak, F., 2017. Static analysis of functionally graded carbon nanotube-reinforced plate and shell structures. Composite Structures, 176, pp.1107-1123.
[15] Zghal, S., Frikha, A. & Dammak, F., 2018. Mechanical buckling analysis of functionally graded power-based and carbon nanotubes-reinforced composite plates and curved panels. Composites Part B: Engineering, 150, pp.165-183.
[16] Frikha, A., Zghal, S. & Dammak, F., 2018. Dynamic analysis of functionally graded carbon nanotubes-reinforced plate and shell structures using a double directors finite shell element. Aerospace Science and Technology, 78, pp.438-451.
[17] Zghal, S., Frikha, A. & Dammak, F., 2018. Non-linear bending analysis of nanocomposites reinforced by graphene-nanotubes with finite shell element and membrane enhancement. Engineering Structures, 158, pp.95-109.
[18] Mehar, K. & Panda, S.K., 2019. Theoretical deflection analysis of multi-walled carbon nanotube reinforced sandwich panel and experimental verification. Composites Part B: Engineering, 167, pp.317-328.
[19] Mindlin, R.D., 1965. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1 (4), pp.417-438.
[20] Aifantis, E.C., 1999. Strain gradient interpretation of size effects. International Journal of Fracture, 95 (1), pp.299-314.
[21] Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. & Tong, P., 2003. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51 (8), pp.1477-1508.
[22] Lim, C.W., Zhang, G. & Reddy, J.N., 2015. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, pp.298-313.
[23] Gui, Y. & Wu, R., 2023. Buckling analysis of embedded thermo-magneto-electro-elastic nano cylindrical shell subjected to axial load with nonlocal strain gradient theory. Mechanics Research Communications, 128, pp.104043.
[24] Lu, L., Guo, X. & Zhao, J., 2019. A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects. Applied Mathematical Modelling, 68, pp.583-602.
[25] Arefi, M., Kiani, M. & Rabczuk, T., 2019. Application of nonlocal strain gradient theory to size dependent bending analysis of a sandwich porous nanoplate integrated with piezomagnetic face-sheets. Composites Part B: Engineering, 168, pp.320-333.
[26] Farajpour, A., Yazdi, M.R.H., Rastgoo, A. & Mohammadi, M., 2016. A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment. Acta Mechanica, 227 (7), pp.1849-1867.
[27] Nematollahi, M.S. & Mohammadi, H., 2019. Geometrically nonlinear vibration analysis of sandwich nanoplates based on higher-order nonlocal strain gradient theory. International Journal of Mechanical Sciences, 156, pp.31-45.
[28] Thai, C.H., Hung, P.T., Nguyen-Xuan, H. & Phung-Van, P., 2023. A size-dependent meshfree approach for magneto-electro-elastic functionally graded nanoplates based on nonlocal strain gradient theory. Engineering Structures, 292, pp.116521.
[29] Kumar, D., Ali, S.F. & Arockiarajan, A., 2021. Theoretical and experimental studies on large deflection analysis of double corrugated cantilever structures. International Journal of Solids and Structures, 228, pp.111126.
[30] Han, P., Tian, F. & Babaei, H., 2023. Thermally induced large deflection analysis of graphene platelet reinforced nanocomposite cylindrical panels. Structures, 53, pp.1046-1056.
[31] Bertóti, E., 2020. Primal- and dual-mixed finite element models for geometrically nonlinear shear-deformable beams – a comparative study. Computer Assisted Methods in Engineering and Science, 27 (4), pp.285-315.
[32] Liu, L., Zhong, X. & Liao, S., 2023. Accurate solutions of a thin rectangular plate deflection under large uniform loading. Applied Mathematical Modelling, 123, pp.241-258.
[33] Wang, J. & Xiao, J., 2022. Analytical solutions of bending analysis and vibration of rectangular nano laminates with surface effects. Applied Mathematical Modelling, 110, pp.663-673.
[34] Mehar, K., Panda, S.K. & Mahapatra, T.R., 2018. Thermoelastic deflection responses of cnt reinforced sandwich shell structure using finite element method. Scientia Iranica, 25 (5), pp.2722-2737.
[35] Mehar, K., Panda, S.K. & Patle, B.K., 2018. Stress, deflection, and frequency analysis of cnt reinforced graded sandwich plate under uniform and linear thermal environment: A finite element approach. Polymer Composites, 39 (10), pp.3792-3809.
[36] Mehar, K. & Panda Subrata, K., 2017. Nonlinear static behavior of fg-cnt reinforced composite flat panel under thermomechanical load. Journal of Aerospace Engineering, 30 (3), pp.04016100.
[37] Mehar, K. & Panda, S.K., 2018. Elastic bending and stress analysis of carbon nanotube-reinforced composite plate: Experimental, numerical, and simulation. Advances in Polymer Technology, 37 (6), pp.1643-1657.
[38] Mehar, K., Panda, S.K. & Mahapatra, T.R., 2018. Large deformation bending responses of nanotube-reinforced polymer composite panel structure: Numerical and experimental analyses. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 233 (5), pp.1695-1704.
[39] Ambartsumian, S.A., 1960. On the theory of bending of anisotropic plates and shallow shells. Journal of Applied Mathematics and Mechanics, 24 (2), pp.500-514.
[40] Reddy, J.N., 1984. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics, 51 (4), pp.745-752.
[41] Reissner, E., 1975. On transverse bending of plates, including the effect of transverse shear deformation. International Journal of Solids and Structures, 11 (5), pp.569-573.
[42] Touratier, M., 1991. An efficient standard plate theory. International Journal of Engineering Science, 29 (8), pp.901-916.
[43] Soldatos, K.P., 1992. A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mechanica, 94 (3), pp.195-220.
[44] Aydogdu, M., 2009. A new shear deformation theory for laminated composite plates. Composite Structures, 89 (1), pp.94-101.
[45] Mantari, J.L., Oktem, A.S. & Guedes Soares, C., 2012. A new higher order shear deformation theory for sandwich and composite laminated plates. Composites Part B: Engineering, 43 (3), pp.1489-1499.
[46] Li, Q., Wu, D., Chen, X., Liu, L., Yu, Y. & Gao, W., 2018. Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on winkler–pasternak elastic foundation. International Journal of Mechanical Sciences, 148, pp.596-610.
[47] Kerr, A.D. & Alexander, H., 1968. An application of the extended kantorovich method to the stress analysis of a clamped rectangular plate. Acta Mechanica, 6 (2), pp.180-196.
[48] Bellman, R. & Casti, J., 1971. Differential quadrature and long-term integration. Journal of Mathematical Analysis and Applications, 34 (2), pp.235-238.
[49] Mousavi, S.M. & Tahani, M., 2012. Analytical solution for bending of moderately thick radially functionally graded sector plates with general boundary conditions using multi-term extended kantorovich method. Composites Part B: Engineering, 43 (3), pp.1405-1416.
Mechanics of Advanced Composite Structures
Volume 13, Issue 1 - Serial Number 27
In Progress
April 2026
Pages 117-128

Files

Share

How to cite

Statistics