Derivation of Nonlocal Finite Element Formulation for Nano Beams

Year 2018, Volume: 10 Issue: 2, 131 - 139, 15.08.2018

Ömer Civalek , Hayri Metin Numanoglu , Büşra Uzun

https://doi.org/10.24107/ijeas.450239

Cited By: 1

Abstract

In the
present paper, a new nonlocal formulation for vibration derived for nano beam lying
on elastic matrix. The formulation is based on the cubic shape polynomial
functions via finite element method. The size effect on finite element matrix
is investigated using nonlocal elasticity theory. Finite element formulations
and matrix coefficients have been obtained for nano beams. Size-dependent stiffness
and mass matrix are derived for Euler-Bernoulli beams. 

Keywords

Size-depended vibration, cubic shape functions, Nonlocal elasticity, Euler-Bernoulli beam, new mass and stiffness matrix.

References

• Sun, C. T., and Haitao, Z., Size-dependent elastic moduli of platelike nanomaterials. Journal of Applied Physics 93(2), 1212-1218, 2003.
• Zhu, R., Pan, E., and Roy, A. K., Molecular dynamics study of the stress–strain behavior of carbon-nanotube reinforced Epon 862 composites. Materials Science and Engineering: A, 447(1), 51-57. 2007.
• Liang, Y. C., Dou, J. H., and Bai, Q. S., Molecular dynamic simulation study of AFM single-wall carbon nanotube tip-surface interactions. In Key Engineering Materials, 339, 206-210, 2007.
• Fleck, N. A., and Hutchinson, J. W., Strain gradient plasticity. Advances in applied mechanics, 33, 296-361, 1997.Hadjesfandiari, A. R., and Dargush, G. F., Couple stress theory for solids. International Journal of Solids and Structures, 48, 2496–2510, 2011.
• Yang, F. A. C. M., Chong, A. C. M., Lam, D. C. C., & Tong, P., Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), 2731-2743, 2002.
• Ma, H. M., Gao, X. L., Reddy, J. N., A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of Mechanics Physics and Solids, 56, 3379–3391, 2008.
• Reddy, J. N., Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids, 59, 2382–2399, 2011.
• Zhou, S. J., & Li, Z. Q., Length scales in the static and dynamic torsion of a circular cylindrical micro-bar. Journal of Shandong university of technology, 31(5), 401-407, 2001.
• Akgöz, B., Civalek, O., Longitudinal vibration analysis for microbars based on strain gradient elasticity theory. Journal of Vibration and Control, 20(4), 606-616, 2014.
• Asghari, M., Kahrobaiyan, M. H., Ahmadian, M. T., A nonlinear Timoshenko beam formulation based on the modified couple stress theory. International Journal of Engineering Science, 48(12), 1749-1761, 2010.
• Akgoz, B., Civalek, O., Shear deformation beam models for functionally graded microbeams with new shear correction factors. Composite Structures 112, 214-225, 2014.
• Akgöz, B., Civalek, O., Buckling analysis of cantilever carbon nanotubes using the strain gradient elasticity and modified couple stress theories. Journal of Computational and Theoretical Nanoscience, 8(9), 1821-1827, 2011.Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of applied physics, 54(9), 4703-4710, 1983.
• Eringen, A. C., Nonlocal continuum field theories. Springer Science & Business Media, 2002.
• Chen, Y., Lee, J. D., Eskandarian, A., Atomistic viewpoint of the applicability of microcontinuum theories. International journal of solids and structures, 41(8), 2085-2097, 2004.
• Peddieson, J., Buchanan, G. R., McNitt, R. P., Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science, 41(3), 305-312, 2003.
• Sudak, L.J., Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. Journal of Applied Physics, 94, 7281–7287, 2003.
• Ansari, R., Rajabiehfard, R., Arash, B., Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets. Computational Materials Science, 49(4), 831-838, 2010.
• Mahmoud, F. F., Eltaher, M. A., Alshorbagy, A. E., & Meletis, E. I., Static analysis of nanobeams including surface effects by nonlocal finite element. Journal of Mechanical Science and Technology, 26(11), 3555-3563, 2012.
• Pradhan, S. C., Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory. Finite Elements in Analysis and Design, 50, 8-20, 2012.
• Phadikar, J. K., Pradhan, S. C., Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Computational materials science, 49(3), 492-499, 2010.
• Eltaher, M. A., Alshorbagy, A. E., Mahmoud, F. F., Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 37(7), 4787-4797, 2013.
• Demir, Ç., Civalek, O., Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Applied Mathematical Modelling, 37(22), 9355-9367, 2013.
• Civalek, O., Demir, C., A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation, 289, 335-352, 2016.
• Ansari, R., Gholami, R., Hosseini, K., Sahmani, S., A sixth-order compact finite difference method for vibrational analysis of nanobeams embedded in an elastic medium based on nonlocal beam theory. Mathematical and Computer Modelling, 54(11), 2577-2586, 2011.
• Karimi, M., Shahidi, A. R., Finite difference method for sixth-order derivatives of differential equations in buckling of nanoplates due to coupled surface energy and non-local elasticity theories. International Journal of Nano Dimension, 6(5), 525, 2015.
• Pradhan, S. C., Reddy, G. K., Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM. Computational Materials Science, 50(3), 1052-1056, 2011.
• Senthilkumar, V., Buckling analysis of a single-walled carbon nanotube with nonlocal continuum elasticity by using differential transform method. Advanced Science Letters, 3(3), 337-340, 2010.
• Ebrahimi, F., Salari, E., Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method. Composites Part B: Engineering, 79, 156-169, 2015.
• Pradhan, S. C., Kumar, A., Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Composite Structures, 93(2), 774-779, 2011.
• Pradhan, S. C., Kumar, A., Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method. Computational Materials Science, 50(1), 239-245, 2010.
• Danesh, M., Farajpour, A., Mohammadi, M., Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method. Mechanics Research Communications, 39(1), 23-27, 2012.
• Murmu, T., Pradhan, S. C., Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory. Computational Materials Science, 46(4), 854-859, 2009.
• Civalek, O., Demir, Ç., Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory. Applied Mathematical Modeling, 35, 2053-2067, 2011
• Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, 288- 307, 2007.
• Reddy, J. N., Pang, S. D., Nonlocal continuum theories of beams for the analysis of carbon nanotubes. Journal of Applied Physics, 103(2), 023511, 2008.
• Mercan, K., Civalek, O.,  DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix, Composite Structures, 143, 300-309, 2016.
• Baltacıoglu, A.K., Akgoz, B., Civalek, O., Nonlinear static response of laminated composite plates by discrete singular convolution method, Composite Structures, 93, 153–161, 2010.
• Civalek, O., Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Engineering Structures, 26, 171–186, 2004.
• Demir, Ç., Mercan, K., Civalek, O., Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel, Composites Part B: Engineering, 94, 1-10, 2016.
• Civalek, O., Finite Element analysis of plates and shells. Elazığ: Fırat University, 1998.
• Mercan, K., Ö Civalek, O., Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ, Composites Part B: Engineering, 114, 34-45, 2017.