Effect of Rotary Inertia on Vibrational Response of Embedded Graphene Sheets

Year 2016, , 75 - 86, 29.12.2016

Bekir Akgöz

https://doi.org/10.24107/ijeas.279739

Abstract

In the present study, influence of rotary inertia on the size-dependent
free vibration analysis of embedded single-layered graphene sheets is examined
based on modified couple stress theory. Governing differential equations and
corresponding boundary conditions in motion are derived by implementing
Hamilton’s principle on the basis of Kirchhoff thin plate theory. Also, effect
of elastic foundation is taken into consideration by using a two-parameter
elastic foundation model. Influences of additional material length scale
parameter, mode number, elastic foundation and rotary inertia on the natural
frequencies are investigated in detail.

Keywords

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

References

• [1] Iijima, S. Helical Microtubules of Graphitic Carbon. Nature, 354, 56–58, 1991.
• [2] Poole, W.J., Ashby, M.F., Fleck, N.A. Micro-hardness of annealed and work- hardened copper polycrystals. Scripta Mater., 34, 559–564, 1996.
• [3] Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P. Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids, 51, 1477–1508, 2003.
• [4] McFarland, A.W., Colton, J.S. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng., 15, 1060–1067, 2005.
• [5] Reddy, J.N., Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, 2nd Edition, 2002.
• [6] Soedel,W., Vibrations of shells and plates, CRC Press, 3rd Edition, 2004.
• [7] Viola, E., Tornabene, F., Fantuzzi, N. Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories. Compos. Struct., 101, 59–93, 2013.
• [8] Ferreira, A.J.M., Viola, E., Tornabene, F., Fantuzzi, N., Zenkour, A.M. Analysis of sandwich plates by generalized differential quadrature method. Math. Probl. Eng., 964367, 2013.
• [9] Civalek, Ö. Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches. Compos. Part B: Eng., 50, 171–179, 2013.
• [10] Tornabene, F., Fantuzzi, N., Bacciocchi, M., Dimitri, R. Free vibrations of composite oval and elliptic cylinders by the generalized differential quadrature method. Thin-Walled Struct., 97, 114–129, 2015.
• [11] Pradhan, S.C., Loy, C.T., Lam, K.Y., Reddy, J.N. Vibration characteristics of functionally graded cylindrical shells under various boundary conditions. Appl. Acoustics, 61, 111–124, 2000.
• [12] 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. Vessel. Pip., 88, 290–300, 2011.
• [13] Civalek, Ö., Korkmaz, A., Demir, Ç. Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges. Adv. Eng. Softw., 41, 557–560, 2010.
• [14] Akgöz, B., Civalek, Ö. Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations. Steel and Compos. Struct., 11, 403–421, 2011.
• [15] Civalek, Ö. Analysis of thick rectangular plates with symmetric cross-ply laminates based on first-order shear deformation theory. J. Compos. Mater., 42, 2853–2867, 2008.
• [16] Gürses, M., Civalek, Ö., Korkmaz, A., Ersoy, H. Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first‐order shear deformation theory. Int. J. Numer. Meth. Eng., 79, 290–313, 2009.
• [17] Baltacıoğlu, A.K., Akgöz, B., Civalek, Ö. Nonlinear static response of laminated composite plates by discrete singular convolution method. Compos. Struct., 93, 153–161, 2010.
• [67] Ma, H.M., Gao, X.L., Reddy, J.N. A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech., 220, 217–235, 2011.
• [19] Mercan, K., Demir, Ç., Civalek, Ö. Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique. Curved and Layer. Struct., 3, 82–90, 2016.
• [20] Avcar, M. Elastic buckling of steel columns under axial compression. American J. Civ. Eng., 2, 102–108, 2014.
• [21] Avcar, M. Effects of rotary inertia shear deformation and non-homogeneity on frequencies of beam. Struct. Eng. Mech., 55, 871–884, 2015.
• [22] Emsen, E., Mercan, K., Akgöz, B., Civalek, Ö. Modal Analysis of Tapered Beam-Column Embedded in Winkler Elastic Foundation. Int. J. Eng. Appl. Sci., 7, 25–35, 2015.
• [23] Mindlin, R.D., Tiersten, H.F. Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal., 11, 415–448, 1962.
• [24] Koiter, W.T. Couple stresses in the theory of elasticity: I and II. Proc. K Ned. Akad. Wet. (B), 67, 17–44, 1964.
• [25] Toupin, R.A. Theory of elasticity with couple stresses. Arch. Ration. Mech. Anal., 17, 85–112, 1964.
• [26] Eringen, A.C. Theory of micropolar plates. Z .Angew. Math. Phys., 18, 12–30, 1967.
• [27] Eringen, A.C. Nonlocal polar elastic continua. Int. J. Eng. Sci., 10, 1–16, 1972.
• [28] Eringen, A.C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys., 54, 4703–4710, 1983.
• [29] Vardoulakis, I., Sulem, J. Bifurcation Analysis in Geomechanics. London: Blackie/Chapman & Hall, 1995.
• [30] Aifantis, E.C. Gradient deformation models at nano, micro, and macro scales. J. Eng. Mater. Technol., 121, 189–202, 1999.
• [31] Fleck, N.A., Hutchinson, J.W. A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids, 41, 1825–1857, 1993.
• [32] Fleck, N.A., Hutchinson, J.W. A reformulation of strain gradient plasticity. J. Mech. Phys. Solids, 49, 2245–2271, 2001.
• [33] Kong, S., Zhou, S., Nie, Z., Wang, K. Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int. J. Eng. Sci., 47, 487–498, 2009.
• [34] Wang, B., Zhao, J., Zhou, S. A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur. J. Mech. A/Solids, 29, 591-599, 2010.
• [35] 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.
• [36] Akgöz, B., Civalek, Ö. Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Arch. Appl. Mech., 82, 423–443, 2012.
• [37] Kahrobaiyan, M.H., Rahaeifard, M., Tajalli, S.A., Ahmadian, M.T. A strain gradient functionally graded Euler-Bernoulli beam formulation. Int. J. Eng. Sci., 52, 65–76, 2012.
• [38] Akgöz, B., Civalek, Ö. Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mech., 224, 2185–2201, 2013.
• [39] Ansari, R., Gholami, R., Sahmani, S. Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Compos. Struct., 94, 221–228, 2011.
• [40] Akgöz, B., Civalek, Ö. Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Compos. Struct., 134, 294–301, 2015.
• [41] Ghayesh, M.H., Amabili, M., Farokhi, H. Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci., 63, 52–60, 2013.
• [42] Akgöz, B., Civalek, Ö. Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity. Struct. Eng. Mech., 48, 195–205, 2013.
• [43] Akgöz, B., Civalek, Ö. Investigation of size effects on static response of single-walled carbon nanotubes based on strain gradient elasticity. Int. J. Comput. Methods, 9, 1240032, 2012.
• [44] Kahrobaiyan, M.H., Asghari, M., Ahmadian, M.T. Strain gradient beam element. Finite Elem. Anal. Des., 68, 63–75, 2013.
• [45] Akgöz, B., Civalek, Ö. Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronaut., 119, 1–12, 2016.
• [46] Akgöz, B., Civalek, Ö. A novel microstructure-dependent shear deformable beam model. Int. J. Mech. Sci., 99, 10–20, 2015.
• [47] Zhang, B., He, Y., Liu, D., Gan, Z., Shen, L. Non-classical Timoshenko beam element based on the strain gradient elasticity theory. Finite Elem. Anal. Des., 79, 22–39, 2014.
• [48] Akgöz, B., Civalek, Ö. A new trigonometric beam model for buckling of strain gradient microbeams. Int. J. Mech. Sci., 81, 88–94, 2014.
• [49] Lei, J., He, Y., Zhang, B., Gan, Z., Zeng, P. Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory. Int. J. Eng. Sci., 72, 36–52, 2013.
• [50] Akgöz, B., Civalek, Ö. A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mech., 226, 2277–2294, 2015.
• [51] Wang, B., Zhou, S., Zhao, J., Chen, X. A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory. Eur. J. Mech. A Solids, 30, 517–524, 2011.
• [52] Movassagh, A.A., Mahmoodi, M.J. A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory. Eur. J. Mech. A Solids, 40, 50–59, 2013.
• [53] Li, A., Zhou, S., Zhou, S., Wang, B. A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory. Compos. Struct., 113, 272–280, 2014.
• [54] Sahmani, S., Ansari, R. On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory. Compos. Struct., 95, 430–442, 2013.
• [55] Ansari, R., Gholami, R., Faghih Shojaei, M., Mohammadi, V., Sahmani, S. Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. Eur. J. Mech. A Solids, 49, 251–267, 2015.
• [56] Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P. Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct., 39, 2731–2743, 2002.
• [57] Park, S.K., Gao, X.L. Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng., 16, 2355–2359, 2006.
• [58] Kong, S., Zhou, S., Nie, Z., Wang, K. The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int. J. Eng. Sci., 46, 427–437, 2008.
• [59] Ma, H.M., Gao, X.L., Reddy, J.N. A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids, 56, 3379–3391, 2008.
• [60] Reddy, J.N. Microstructure-dependent couples stress theories of functionally graded beams. J. Mech. Phys. Solids, 59, 2382–2399, 2011.
• [61] Akgöz, B., Civalek, Ö. Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Compos. Struct., 98, 314–322, 2013.
• [62] Akgöz, B., Civalek, Ö. Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory. Mater. Des., 42, 164–171, 2012.
• [63] 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.
• [64] Gao, X.-L., Huang, J.X., Reddy, J.N. A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech., 224, 2699–2718, 2013.
• [65] Jomehzadeh, E., Noori, H.R., Saidi, A.R. The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Phys. E, 43, 877–883, 2011.
• [66] Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S. Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory. J. Sound Vib., 331, 94–106, 2012.
• [86] Arda, M., Aydogdu, M. Bending of CNTs under the Partial Uniform Load. Int. J. Eng. Appl. Sci., 8, 21–29, 2016.
• [69] Thai, H.T., Kim, S.E. A size-dependent functionally graded Reddy plate model based on a modified couple stress theory. Compos. Part B: Eng., 45, 1636–1645, 2013.
• [70] Thai, H.T., Vo, T.P. A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory. Compos. Struct., 96, 376–383, 2013.
• [71] Tsiatas, G.C. A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct., 46, 2757–2764, 2009.
• [72] Yin, L., Qian, Q., Wang, L., Xia, W. Vibration analysis of microscale plates based on modified couple stress theory. Acta Mech. Solida Sin., 23, 386–393, 2010.
• [73] Peddieson, J., Buchanan, G.R., McNitt, R.P. Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci., 41, 305–312, 2003.
• [74] Civalek, Ö., Demir, Ç. A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl. Math. Comput., 289, 335–352, 2016.
• [75] Wang, Q., Liew, K.M. Application of nonlocal continuum mechanics to static analysis of micro and nano-structures. Phys. Lett. A, 363, 236–242, 2007.
• [76] Civalek, Ö., Demir, Ç., Akgöz, B. Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory. Int. J. Eng. Appl. Sci., 1, 47–56, 2009.
• [77] Demir, Ç., Civalek, Ö. Nonlocal Finite Element Formulation for Vibration. Int. J. Eng. Appl. Sci., 8, 109–117, 2016.
• [78] Reddy, J.N. Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci., 45, 288–307, 2007.
• [79] Thai, H.T. A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci., 52, 56–64, 2012.
• [80] Civalek, Ö., Akgöz, B. Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix. Comput. Mater. Sci., 77, 295–303, 2013.
• [83] Demir, Ç., Civalek, Ö. Nonlocal Deflection of Microtubules Under Point Load, Int. J. Eng. Appl. Sci., 7, 33–39, 2015.
• [82] Aydogdu, M. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys. E, 41, 1651–1655, 2009.
• [84] Mercan, K., Civalek, Ö. Buckling Analysis of Silicon Carbide Nanotubes (SiCNTs). Int. J. Eng. Appl. Sci., 8, 101–108, 2016.
• [85] Civalek, Ö., Mercan, K., Demir, Ç., Akgöz, B. Coordinate Transformation for Sector and Annular Sector Shaped Graphene Sheets on Silicone Matrix. Int. J. Eng. Appl. Sci., 7, 56–73, 2015.