Coordinate Transformation for Sector and Annular Sector Shaped Graphene Sheets on Silicone Matrix

Year 2015, Volume: 7 Issue: 2, 56 - 73, 01.06.2015

Kadir Mercan , Çiğdem Demir Bekir Akgöz Ömer Civalek

https://doi.org/10.24107/ijeas.251247

Cited By: 8

Abstract

In the present manuscript, we developed a systematic formulation for some type graphene sheets having annular sector, sector shaped or curvilinear side graphene located on a silicone matrix via nonlocal elasticity theory for numerical solution. An eight-node curvilinear element is used for transformation of the governing equation of motion of annular sector graphene from physical region to computational region in conjunctions with the thin plate theory. Silicone matrix is modeled by using the Winkler-Pasternak elastic foundations. The formulation is usefully for different shaped graphene sheets

Keywords

Scale effect , silicone matrix , sector graphene , discrete singular convolution , vibration

References

• [1] Katsnelson, M.I., Graphene: carbon in two dimensions, Cambridge University Press, UK, 2012.
• [2] Van Lier, G., Van Alsenoy, C., Van Doren V., Ab initio study of the elastic properties of single-walled carbon nanotubes and graphene. Chem. Phys. Lett., 326, 181-185, 2000.
• [3] Liu, Y., Xie, B., Zhang, Z., Zheng, Q., Xu, Z., Mechanical properties of graphene papers. J. Mech. Phys. Solids, 60, 591-605, 2012. [4] Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Two-dimensional gas of massless Dirac fermions in graphene. Nature, 438, 197-200, 2005.
• [5] Ohta, T., Bostwick, A., Seyller, T., Horn, K., Rotenberg, E., Controlling the Electronic Structure of Bilayer Graphene. Science, 313, 951-954, 2006.
• [6] Sakhaee-Pour, A., Ahmadian, M.T., Vafai, A., Potential application of single-layered graphene sheet as strain sensor. Solid State Commun., 147, 336-340, 2008.
• [7] Krishnan, A., Dujardin, E., Ebbesen, T.W., Yianilos, P.N., Treacy, M.M.J., Young's modulus of single-walled nanotubes. Phys. Rev. Lett. B, 58, 14013-14019, 1998.
• [8] He, X.Q., Wang, J.B., Liu, B., Liew, K.M., Analysis of nonlinear forced vibration of multi-layered graphene sheets. Comput. Mater. Sci., 61, 194-199, 2012.
• [9] Shen, Z.B., Tang, H.L., Li, D.K., Tang, G.J., Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory. Comput. Mater. Sci., 61, 200-205, 2012.
• [10] Baykasoglu, C., Mugan, A., Dynamic analysis of single-layer graphene sheets. Comput. Mater. Sci., 55, 228-236, 2012.
• [11] Lin, R.M., Nanoscale vibration characterization of multi-layered graphene sheets embedded in an elastic medium. Comput. Mater. Sci., 53, 44-52, 2012.
• [12] Reddy, J.N., Pang, S.D., Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J. Appl. Phy., 103, 023511, 2008.
• [13] Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys., 54, 4703-4710, 1983.
• [14] Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci., 45, 288-307, 2007.
• [15] Ansari, R., Sahmani, S., Arash, B., Nonlocal plate model for free vibrations of single-layered graphene sheets. Phys. Lett. A, 375, 53-62, 2010.
• [16] Ansari R, Arash B., Rouhi H. Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity. Compos. Struct., 93, 2419-2429, 2011.
• [17] Xie, G.Q., Han, X., Liu, G.R., Long, S.Y., Effect of small size-scale on the radial buckling pressure of a simply supported multi-walled carbon nanotube. Smart Mater. Struct., 15(4), 1143-1149, 2006.
• [18] Malekzadeh, P., Setoodeh, A.R., Alibeygi Beni, A., Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates. Compos. Struct., 93, 1631-1639, 2011.
• [19] Malekzadeh, P., Setoodeh, A.R., Alibeygi Beni, A., Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium. Compos. Struct., 93, 2083-2089, 2011.
• [20] Aghababaei, R., Reddy, J.N., Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J. Sound Vib., 326, 277-289, 2009.
• [21] Lu, P., Zhang, P.Q., Lee, H.P., Wang, C.M., Reddy, J.N., Non-local elastic plate theories. Proc. R. Soc. A., 463, 3225-3240, 2007.
• [22] Civalek, Ö., Demir, Ç., Bending analysis of microtubules using nonlocal EulerBernoulli beam theory. Appl. Math. Model., 35(5), 2053-2067, 2011.
• [23] Demir, Ç., Civalek, Ö. and Akgöz, B., Free Vibration Analysis of Carbon Nanotubes Based on Shear Deformable Beam Theory by Discrete Singular Convolution Technique. Math. Comput. Appl., 15, 57-65, 2010.
• [24] Civalek, Ö., Demir, Ç. and Akgöz, B., Free Vibration and Bending Analyses of Cantilever Microtubules Based On Nonlocal Continuum Model. Math. Comput. Appl., 15, 289-298, 2010.
• [25] Civalek, Ö., Akgöz, B., Free vibration analysis of microtubules as cytoskeleton components: Nonlocal Euler-Bernoulli beam modeling. Scientia Iranica., Trans. BMech. Eng., 17(5), 367-375, 2010.
• [26] Mohammadi, M., Ghayour, M., Farajpour, A., Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model. Compos. Part B, 45, 32-42, 2013.
• [27] Akgöz, B., Civalek, Ö., Application of strain gradient elasticity theory for buckling analysis of protein microtubules. Curr. Appl. Phys., 11, 1133-1138, 2011.
• [28] Akgöz, B., Civalek, Ö., Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int. J. Eng. Sci., 49, 1268- 1280, 2011.
• [29] Wei, G.W., Discrete singular convolution for the solution of the Fokker-Planck equations. J. Chem Phys, 110, 8930-8942, 1999.
• [30] Wei, G.W., A new algorithm for solving some mechanical problems. Comput. Methods Appl. Mech. Eng., 190, 2017-2030, 2001.
• [31] Wei, G.W., Vibration analysis by discrete singular convolution. J. Sound Vib. 244, 535-553, 2001.
• [32] Wei, G.W., Discrete singular convolution for beam analysis. Eng. Struct., 23,1045- 1053, 2001,
• [33] Wei, G.W., Zhao Y.B., and Xiang, Y., Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: Theory and algorithm. Int. J. Numer. Methods Eng., 55, 913-946, 2002.
• [34] Wei, G.W., Zhao Y.B., and Xiang, Y., A novel approach for the analysis of highfrequency vibrations. J. Sound and Vib., 257, 207-246, 2002.
• [35] Wei, G.W., Zhao Y.B., and Xiang, Y., The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution. Int. J. Mech. Sci., 43, 1731-1746, 2001.
• [36] Xiang, Y., Zhao, Y.B., and Wei, G.W., Discrete singular convolution and its application to the analysis of plates with internal supports. Part 2: Applications. Int. J. Numer. Methods Eng., 55, 947-971, 2002.
• [37] Zhao, Y.B., Wei G.W, and Xiang, Y., Discrete singular convolution for the prediction of high frequency vibration of plates. Int. J. Solids Struct., 39, 65-88, 2002.
• [38] Zhao, Y.B., and Wei G.W., DSC analysis of rectangular plates with non-uniform boundary conditions. J. Sound Vib., 255(2), 203-228, 2002.
• [39] Zhao, S., Wei, G.W., and Xiang, Y., DSC analysis of free-edged beams by an iteratively matched boundary method. J. Sound Vib., 284, 487-493, 2005.
• [40] Civalek, Ö., Free vibration and buckling analyses of composite plates with straightsided quadrilateral domain based on DSC approach. Finite Elem. Anal. Des., 43, 1013- 1022, 2007.
• [41] Civalek Ö., A parametric study of the free vibration analysis of rotating laminated cylindrical shells using the method of discrete singular convolution. Thin-Walled Struct., 45, 692-698, 2007.
• [42] Civalek Ö., Linear vibration analysis of isotropic conical shells by discrete singular convolution (DSC). Struct. Eng. Mech., 25, 127-130, 2007.
• [43] Civalek, Ö., A four-node discrete singular convolution for geometric transformation and its application to numerical solution of vibration problem of arbitrary straightsided quadrilateral plates. Appl. Math. Model., 33(1), 300-314, 2009.
• [44] Civalek, Ö., Use of Eight-Node Curvilinear Domains in Discrete Singular Convolution Method for Free Vibration Analysis of Annular Sector Plates with Simply Supported Radial Edges. J. Vib. Control, 16, 303-320, 2010.
• [45] Civalek, Ö., Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method, Appl. Math. Model., 33(10), 3825-3835, 2009.
• [46] Civalek, Ö., Free vibration analysis of symmetrically laminated composite plates with first-order shear deformation theory (FSDT) by discrete singular convolution method. Finite Elem. Anal. Des., 44(12-13), 725-731, 2008.
• [47] Civalek, Ö. Geometrically non-linear static and dynamic analysis of plates and shells resting on elastic foundation by the method of polynomial differential quadrature (PDQ), Ph. D. Thesis, Firat University, Elazig, 2004 (in Turkish), 2004.
• [48] Civalek, Ö. Finite Element analysis of plates and shells, Elazığ: Fırat University, 1998.
• [49] Civalek, Ö., The determination of frequencies of laminated conical shells via the discrete singular convolution method. J. Mech. Mater. Struct., 1(1), 163-182, 2006.
• [50] Baltacıoğlu, A.K., Civalek, Ö., Akgöz B., Demir F., Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution. Int. J. Press. Vessels Pip., 88 (8), 290-300, 2011.
• [51] Akgöz, B., Civalek, Ö., Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory. Meccanica, 48, 863-873, 2013.
• [52] Civalek, Ö., Akgöz, B., Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix. Comput. Mater. Sci., 77, 295-303, 2013.
• [53] Baltacıoğlu, A.K., Akgöz, B., Civalek, Ö., Nonlinear static response of laminated composite plates by discrete singular convolution method. Compos. Struct., 93(1), 153-161, 2010.