Vibration analysis of multi-directional functionally graded nanoplates with initial geometric imperfection using quasi-3d theory based on an isogeometric approach

Authors

1 Department of mechanical engineering, central Tehran branch, Islamic Azad University, Tehran, Iran

2 Department of Mechanical Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Abstract

In this paper, effects of initial geometric imperfection on free vibration analysis of multi-directional functionally graded nanoplates are studied using an isogeometric approach together with the quasi-3D shear deformation theory. Geometric imperfections may exist inherently in nanoplates or purposely created by researchers. These imperfections strongly affect the plates’ natural frequency. The initial geometric imperfection is considered as an initial curvature and modeled by an analytical function in the governing equations of the nanoplate. The effective material properties distribution is stated based on the Mori-Tanaka scheme, which in addition to the thickness of the plate, also changes in the in-plane directions. A four-variable quasi-3D theory with new distribution function is proposed. Based on Hamilton’s principle, a weak form of free vibration problem for nonlocal plates is derived. These discrete systems of equations are solved using an isogeometric approach. The accuracy of the present study was verified by comparing the results with those given in published papers. Present results indicate the importance of various parameters, especially imperfection amplitude, and material indexes in all directions, on the free vibration behavior of FG nanoplates.

Keywords


[1] Shahverdi H, Barati MR (2017) Vibration analysis of porous functionally graded nanoplates. Int J Eng Sci 120: 82-99.
[2] Daneshmehr A, Rajabpoor A, Hadi A (2015) Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories. Int J Eng Sci 95: 23-35.
[3] Lü CF, Lim CW, Chen W Q (2009) Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. Int J Solids Struct 46: 1176-1185.
[4] Eringen AC (2002) Nonlocal continuum field theories. Springer, New York.
[5] Sobhy M, Radwan AF (2017) A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates. Int J Appl Mech 9(1):1750008.
[6] عزیزی ع، ستوده ع (1397) تحلیل خمش و ارتعاش   آزاد نانوورق مدرج تابعی با استفاده از نظریه ورق      مرتبه بالای مثلثاتی. نشریه مهندسی مکانیک امیرکبیر 1050-1039 :(5)50.
[7] Najafizadeh MM, Raki M, Yousefi P (2018) Vibration analysis of FG Nanoplate based on    third-order shear deformation theory (TSDT)      and nonlocal elasticity. J Solid Mech 10(3): 464-475.
[8] Senthilnathan NR, Lim SP, Lee KH, Chow ST (1987) Buckling of shear-deformable plates. AIAA J 25: 1268-1271.
[9] Zenkour AM (2013) A simple four-unknown refined theory for bending analysis of functionally graded plates. Appl Math Model 37: 9041-9051.
[10] Barati MR, Shahverdi H (2016) A four-variable plate theory for thermal vibration of embedded FG nanoplates under non-uniform temperature distributions with different boundary conditions. Struct Eng Mech 60(4): 707-727.
[11] Fung CP, Chen CH (2006) Imperfection sensitivity in the nonlinear vibration of functionally graded plates. Eur J Mech A-Solid 25(3): 425-436.
[12] Jalali SK, Pugno NM, Jomehzadeh E (2016) Influence of out-of-plane defects on vibration analysis of graphene sheets: molecular and continuum approaches. Superlattice Microst 91: 331-344.
[13] Lusk MT, Carr LD (2008) Nanoengineering defect structures on graphene. Phys Rev Lett 100: 175503.
[14] Kitipornchai S, Yang J, Liew KM (2004) Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections. Int J Solids Struct 41: 2235-2257.
[15] Chen CH, Hsu C.Y (2007) Imperfection sensitivity in the nonlinear vibration oscillations of initially stressed plates. Appl Math Comput 190(1): 465-475.
[16] Yang J, Huang XL (2007) Nonlinear transient response of functionally graded plates with general imperfections in thermal environments. Comput Methods Appl Mech Eng 196: 2619-2630.
[17] Gupta A, Talha M (2016) An assessment of a non-polynomial based higher order shear and normal deformation theory for vibration response of gradient plates with initial geometric imperfections. Compos Part B-Eng 107: 141-161.
[18] Gupta A, Talha M (2017) Nonlinear flexural and vibration response of geometrically imperfect gradient plates using hyperbolic higher-order shear and normal deformation theory. Compos Part B-Eng 123: 241-261.
[19] Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194: 4135-   4195.
[20] Tran LV, Thai CH, Le HT, Gan BS, Lee J, Nguyen-Xuan H (2014) Isogeometric analysis of laminated composite plates based on a four-variable refined plate theory. Eng Anal Bound Elem 47: 68-81.
[21] Nguyen NT, Huic D, Lee J, Nguyen-Xuan H (2015) An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Engrg 297: 191-218.
[22] Thai CH, Zenkour AM, Abdel Wahab M, Nguyen-Xuan H (2016) A simple four-unknown shear      and normal deformations theory for functionally graded isotropic and sandwich plates based           on isogeometric Analysis. Compos Struct 139: 77-95.
[23] Farzam-Rad SA, Hassani B, Karamodin A (2017) Isogeometric analysis of functionally graded plates using a new quasi-3D shear deformation theory based on physical neutral surface. Compos Part B-Eng 108: 174-189.
[24] Nguyen Hoang X, Nguyen Tuan N, Abdel Wahab M, Bordas SPA, Nguyen- Xuan H, Thuc PV (2017) A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory. Comput Meth Appl Mech Eng 313: 904-940.
[25] Phung-Van P, Lieu QX, Nguyen-Xuan H, Abdel-Wahab M (2017) Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Compos Struct 166: 120-135.
[26] Xue Y, Jin G, Ding H, Chen M (2018) Free vibration analysis of in-plane functionally graded plates using a refined plate theory and isogeometric approach. Compos Struct 192:193-205.
[27] Liu Z, Wang C, Duan G, Tan J (2019) A new refined plate theory with isogeometric approach for the static and buckling analysis of functionally graded plates. Int J Mech Sci 161-162: 105036.
[28] Phung-Van P, Thai CH, Nguyen-Xuan H, Abdel-Wahab M (2019) An isogeometric approach of static and free vibration analyses for porous FG nanoplates. Eur J Mech A-Solid 78: 103851.
[29] Lieu QX, Lee S, Kang J, Lee J (2018) Bending and free vibration analyses of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis. Compos Struct 192.
[30] Farzam A, Hassani B (2019) Isogeometric analysis of in-plane functionally graded porous microplates using modified couple stress theory. Aerosp Sci Technol 91: 508-524.
[31] Karamanli A (2020) Size-dependent behaviors of three directional functionally graded shear and normal deformable imperfect microplates. Compos Struct 113076.
[32] Li S, Zheng S, Chen D (2020) Porosity-dependent isogeometric analysis of bi-directional functionally graded plates. Thin Wall Struct 156: 106999.
[33] خورشیدی ک، بخششی ا، قدیریان ح (1395)    بررسی تاثیرات محیط حرارتی بر ارتعاشات آزاد ورق مستطیلی از جنس مواد تابعی مدرج دوبعدی مستقر بر بستر پسترناک. نشریه علمی مکانیک سازه­ها و شاره­ها 147-137 :(3)6.
[34] هاشمی س، جعفری ع ا (1399) تحلیل ارتعاش      آزاد غیرخطی ورق­های مستطیلی از جنس ماده مدرج تابعی دوجهته. نشریه علمی مکانیک سازه­ها و شاره­ها 52-31 :(1)10.
[35] Hashemi S, Jafari AA (2020) Nonlinear Free and Forced vibrations of in-plane bi-directional functionally graded rectangular plate with temperature-dependent properties. Int J Struct Stab Dy 20(8): 2050097
[36] Vel SS, Batra RC (2002) Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIAA J 40: 1421-1433
[37] Mori T,Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21: 571-574.