Bending Analysis of Functionally Graded Nanobeam Using Chebyshev Pseudospectral Method

Year 2021, , 179 - 188, 31.12.2021

Nurettin Şenyer , Nihat Can İbrahim Keles

https://doi.org/10.24107/ijeas.1036951

Abstract

Static performance of functionally graded cantilever nanobeams exposed to lateral and axial loads from the end was examined by applying the Pseudospectral Chebyshev Method. A solution is given for bending analysis using Euler-Bernoulli beam theory. The nonlocal elasticity theory was first introduced by Eringen and is used to represent effect on a small scale. Using the aforementioned theory, the governing differential equations the phenomenon for functionally graded nanobeams are reproduced. It is supposed that the modulus of elasticity of the beam changes exponentially in the x-axis direction, except for the values taken as constant. The exponential change of material properties may not allow analytical problems to be solved with known methods. Therefore, numerical approach is inevitable for the solution of the problem.

Keywords

Nonlocal elasticity theory, nanobeam, Chebyshev pseudospectral method, bending analysis

References

• Aydogdu, M., Taskin, V., Free vibration analysis of functionally graded beams with simply supported edges, Materials & design, 28(5), 1651-1656, 2007.
• Chakraborty, A., Gopalakrishnan, S., Reddy, J.N., A new beam finite element for the analysis of functionally graded materials, International journal of mechanical sciences, 45(3), 519–539, 2003.
• Ke, L. L., Yang, J., Kitipornchai, S., Xiang, Y., Flexural vibration and elastic buckling of a cracked Timoshenko beam made of functionally graded materials. Mechanics of Advanced Materials and Structures, 16(6), 488-502, 2009.
• Li, X.F., A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J Sound Vib., 318(4-5), 1210–1229, 2008.
• Sina, S.A., Navazi, H.M., Haddadpour, H., An analytical method for free vibration analysis of functionally graded beams, Mater Des, 30(3), 741–747, 2009.
• Şimşek, M., Static analysis of a functionally graded beam under a uniformly distributed load by Ritz method, Int J Eng Appl Sci., 1(3), 1–11, 2009.
• Şimşek, M., Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory, Int J Eng Sci., 48(12), 1721–1732, 2010.
• Şimşek, M., Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories, Nucl Eng Des., 240(4), 697–705, 2010.
• Şimşek, M., Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Compos Struct., 92(4), 904–917, 2010.
• Şimşek, M., Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load, Compos Struct., 92(10), 2532–2546, 2010.
• Şimşek, M., Kocatürk T., Akbaş Ş.D., Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory, Compos Struct., 95(1), 740–747, 2013.
• Thai, H.T., Vo, T.P., Bending and free vibration of functionally graded beams using various higher-order shear deformation beam theories, Int J Mech Sci., 62(1), 57–66, 2012.
• Ying, J., Lü, C.F., Chen, W.Q., Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations, Compos Struct., 84(3), 209–219, 2008.
• Lü, C.F., Lim, C.W., Chen, W.Q., Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory, Int J Solids Struct., 46(5), 1176–1185, 2009.
• Zhong, Z., Yu, T., Analytical solution of a cantilever functionally graded beam, Compos Sci Technol., 67(3-4), 481–488, 2007.
• Eringen, A.C., Nonlocal polar elastic continua, Int J Eng Sci., 10(1), 1–16, 1972.
• Eringen, A.C., Edelen, D.G.B., On nonlocal elasticity, Int J Eng Sci., 10(3), 233–248, 1972.
• Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Phys E Low-Dimensional Syst Nanostructures, 41(9), 1651–1655, 2009.
• Liu, T., Hai, M., Zhao, M., Delaminating buckling model based on nonlocal Timoshenko beam theory for microwedge indentation of a film/substrate system, Eng Fract Mech., 75(17), 4909–4919, 2008.
• Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams, Int J Eng Sci., 45(2-8), 288–307, 2007.
• Narendar, S., Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects, Compos Struct., 93(12), 3093–3103, 2011.
• Pradhan, S.C., Phadikar, J.K., Nonlocal elasticity theory for vibration of nanoplates, J Sound Vib., 325(1-2), 206–223, 2009.
• Shen, L.E., Shen, H.S., Zhang, C.L., Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Comput Mater Sci., 48(3), 680–685, 2010.
• Mercan, K., Civalek, Ö., Buckling analysis of silicon carbide nanotubes (SiCNTs), Int J Eng Appl Sci., 8(2), 101–108, 2016.
• Mercan, K., Demir, Ç., Akgöz, B., Civalek, Ö., Coordinate transformation for sector and annular sector shaped graphene sheets on silicone matrix, Int J Eng Appl Sci., 7(2), 56–73, 2015.
• Mercan, K., Civalek, Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix, Compos Struct., 143(1), 300–309, 2016.
• Gürses, M., Akgöz, B., Civalek, Ö., Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation, Appl Math Comput., 219(6), 3226–3340, 2012.
• Yaylı, M.Ö., Buckling analysis of a rotationally restrained single walled carbon nanotube embedded in an elastic medium using nonlocal elasticity, Int J Eng Appl Sci., 8(2), 40–50, 2016.
• Yaylı, M.Ö., An analytical solution for free vibrations of a cantilever nanobeam with a spring mass system, Int J Eng Appl Sci., 7(4), 10–18, 2016.
• Wang, Q., Shindo, Y., Nonlocal continuum models for carbon nanotubes subjected to static loading, J Mech Mater Struct., 1(4), 663–680, 2006.
• Nazmul, I.M., Devnath, I., Exact analytical solutions for bending of bi-directional functionally graded nanobeams by the nonlocal beam theory using the Laplace transform, Forces Mech., 1(1), 100002, 2020.
• Akgöz, B., Civalek, Ö., Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment, Compos Part B Eng., 150(1), 68–77, 2018.
• Dastjerdi, S., Akgöz, B., New static and dynamic analyses of macro and nano FGM plates using exact three-dimensional elasticity in thermal environment, Compos Struct., 192(1), 626–641, 2018.
• 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(2), 1240032, 2012.
• Mercan, K., Numanoglu, H.M., Akgöz, B., Demir, C., Civalek, Ö., Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix, Arch Appl Mech., 87(11), 1797–1814, 2017.
• Civalek, Ö., Kiracioglu, O., Free vibration analysis of Timoshenko beams by DSC method, Int j Numer Method Biomed Eng., 26(12), 1890–1898, 2010.
• Sciarra, F.M.D., Finite element modelling of nonlocal beams, Phys E., 59(1), 144–149, 2014.
• Nguyen, N.T., Kim, N.I., Lee, J., Mixed finite element analysis of nonlocal Euler–Bernoulli nanobeams, Finite Elem Anal Des., 106(1), 65–72, 2015.
• Wang, Q., Wave propagation in carbon nanotubes via nonlocal continuum mechanics, J Appl Phys., 98(12), 124301, 2005.
• Artan, R., Tepe, A., The initial values method for buckling of nonlocal bars with application in nanotechnology, Eur J Mech., 27(3), 469–477, 2008.
• Yaylı, M.Ö., Kandemir, S.Y., Bending analysis of a cantilever Nanobeam with end forces by Laplace transform, Int J Eng Appl Sci., 9(2), 103–111, 2017.
• Nihat, C., Kurgan, N., Hassan, A.H.A., Buckling Analysis of Functionally Graded Plates Using Finite Element Analysis, Int J Eng Appl Sci., 12(1), 43–56, 2020.
• Peddieson, J., Buchanan, G.R., McNitt, R.P., Application of nonlocal continuum models to nanotechnology, Int J Eng Sci., 41(3-5), 305–312, 2003.
• Trefethen, L.N., Spectral methods in MATLAB, volume 10 of Software, Environments, and Tools, Soc Ind Appl Math (SIAM), Philadelphia, PA 2000.
• Fornberg, B., A practical guide to pseudospectral methods, Cambridge university press, 1998.
• Gottlieb, D., The stability of pseudospectral-Chebyshev methods, Math Comput., 36(153), 107–118, 1981.