Assessment of Third‌-order Shear Deformation Graphene Nanoplate Response under Static Loading Using Modified Couple Stress Theory

محورهای موضوعی : MEMS/NEMS

Majid Eskandari Shahraki ¹ (Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran)
Mahmoud Shariati ² (Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran)
Naser Asiaban ³ (Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran)
Ali Davar ⁴ (Faculty of Materials and Manufacturing Technologies, Malek Ashtar University of Technology, Tehran, Iran)
Mohsen Heydari Beni ⁵ (Faculty of Materials and Manufacturing Technologies, Malek Ashtar University of Technology, Tehran, Iran)
Jafar Eskandari Jam ⁶ (Faculty of Materials and Manufacturing Technologies, Malek Ashtar University of Technology, Tehran, Iran)

تاریخ دریافت : 1401/01/02 تاریخ پذیرش : 1401/02/19 تاریخ انتشار : 1400/11/12

کلید واژه: Bending, Buckling, Navier Solution Method, Modified couple-stress theory, Third-order Nano-plate,

چکیده مقاله :

In this paper bending and buckling characteristics of third-order shear, and deformation nanoplates were investigated using the modified couple stress theory and Navier type solution. It can be useful for designing and manufacturing micro-electromechanical and nano-electromechanical systems. The modified couple stress theory was applied to provide the possibility of considering the effects of small scales that have only one material length scale parameter. In this theory, the strain energy density is a function of the strain tensor components, curvature tensor, stress tensor, and the symmetric part of the couple stress tensor. After obtaining the strain energy, external work, and buckling equations, the Hamilton principle is employed to derive the governing equations. Furthermore, by applying boundary and loading conditions in the governing equations, the bending and buckling of a third-order shear deformation nanoplate with simply-supported bearings are obtained and the Navier’s solution is used to solve the equations. The results indicate that the third-order nanoplate subjected to sinusoidal loading yields smaller values of dimensionless bending than it does while subjected to uniform surface traction. It was also found that by increasing the length to thickness ratio, the value of the dimensionless bending of nanoplate decreases but by increasing the aspect ratio of the plate, this value increases. Furthermore, it was shown that the critical buckling load of the third-order nanoplate under uniaxial loading increases by increasing the ratio of the length scale parameter to the thickness of the nanoplate but it decreases by increasing the length to thickness ratio of the nanoplate.

منابع و مأخذ:

[1] Thanh, C.-L., Tran, L. V., Vu-Huu, T. and Abdel-Wahab, M. 2019. The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis. Computer Methods in Applied Mechanics and Engineering. 350: 337-361.

[2] Sladek, V., Sladek, J., Repka, M. and Sator, L. 2020. FGM micro/nano-plates within modified couple stress elasticity. Composite Structures. 245: 112294.

[3] Al-Shewailiah, D. M. R. and Al-Shujairi, M. A. 2021. Static bending of functionally graded single-walled carbon nanotube conjunction with modified couple stress theory.  Materials Today: Proceedings.

[4] Yang, Y., Hu, Z.-L. and Li, X.-F. 2021. Axisymmetric bending and vibration of circular nanoplates with surface stresses. Thin-Walled Structures. 166: 108086.

[5] Aghababaei, R. and Reddy, J.N. 2009. Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. Journal of Sound and Vibration. 326:  277-289.

[6] Jung, W.Y., Han, S.C. and Park, W.T. 2014. A modified couple stress theory for buckling analysis of S-FGM nanoplates embedded in Pasternak elastic medium. Composites Part B: Engineering. 60:  746-756.

[7] Shafiei, Z., Sarrami-Foroushani, S., Azhari, F. and Azhari, M. 2020.  Application of modified couple-stress theory to stability and free vibration analysis of single and multi-layered graphene sheets. Aerospace Science and Technology. 98: 105652.

[8]  Wu, C.-P. and Hu, H.-X. 2021. A unified size-dependent plate theory for static bending and free vibration analyses of micro- and nano-scale plates based on the consistent couple stress theory. Mechanics of Materials. 162: 104085.

[9] Haghshenas, A. and  Shahrajabian, H. 2021. The mechanical properties of PA6/NBR/clay/CaCO3 hybrid nanocomposites. Journal of Modern Processes in Manufacturing and Production. 10(1): 19-30.

[10]  Ameen, S., Akhtar, M. S. and Shin, H. 2020. Graphene Production and Application. IntechOpen. https://doi.org/10.5772/intechopen.83309

[11]  Sun, Y., Sun, M. and  Xie,D. 2018. Graphene Electronic Devices. Academic Press.

[12] Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. 2002. Couple stress Based Strain gradient theory for elasticity. Int.J.Solids Struct. 39:  2731–2743.

[13] Toupin, R.A. 1962. Elastic materials with couple stresses. Arch.Rational Mech.Anal. 11: 385–414.

[14] Mindlin, R.D. and Tiersten, H.F. 1962. Effects of couple-stresses in linear elasticity.  Arch. Rational Mech. Anal. 11: 415–448.

[15] Koiter, W.T. 1964. Couple stresses in the theory of elasticity. I and II.Proc .K. Ned. Akad .Wet.(B) 67: 17–44.

[16] Mindlin, R.D. 1964. Micro-structure in linear elasticity. Arch.RationalMech.Anal. 16: 51–78.

[17] Tsiatas, G.C. 2009. A new kirchhoff  model based on a modified couple stress theory. International Journal of solids and structures. 46: 2757-2764.

[18] Wang, B., Zhou, S., Zhao, J. and Chen, X. 2011. A size-dependent kirchhoff micro-plate model based on strain gradient elasticity theory. European Journal of mechanics A/Solids. 30: 517-524.

[19] Farajpour, A., Shahidi, A.R., Mohammadi, M. and Mahzoon, M. 2012. Buckling of orthotropic micro/nanoscale plates under linearly varying in-plate load via nonlocal continuum mechanics. Composite Structures. 94: 1605-1615.

[20] Tai, T. and HoChoi, D. 2013. size-dependent functionally graded kirchhoff and mindlin plate theory based on a modified couple stress theory. Composite Structures. 95: 142-153.

[21] Akgoz, B. and Civalek, O. 2012. Free vibration analysis for single –layered graphene sheets in an elastic matrix via modified couple stress theory. materials and design. 42: 164-171.

[22] Roque, C.M.C., Ferreira, A.J.M. and Reddy, J.N. 2013. Analysis of mindlin micro plates with a modified couple stress theory and meshlessmethod. Applied Mathematical Modeling. 37: 4626-4633.