Examination of How Size-Effect Modifies the Stiffness and Mass Matrices of Nanotrusses/Nanoframes

Year 2021, Volume: 13 Issue: 4, 155 - 165, 31.12.2021

Hayri Metin Numanoğlu

https://doi.org/10.24107/ijeas.1036574

Cited By: 2

Abstract

The effect of the nonlocal parameter on the free vibration analysis of nano scaled trusses and frames is examined. Accordingly, firstly, the axial and bending vibrations of the nano scaled longitudinal element are formulated. Simple rod and Euler-Bernoulli assumptions are considered for axial and bending vibrations, respectively. Finite element matrices are obtained by applying the average weighted residue to the nonlocal equation of motion for free vibration. These matrices are combined according to the freedoms of longitudinal element and the matrix displacement method is explained for structures consisting of discrete longitudinal elements. It is discussed how the classical stiffness and mass matrices are modified by the atomic parameter.

Keywords

Free vibration, mass matrix, matrix displacement method, nanoframe, nanotruss, nonlocal parameter, stiffness matrix.

References

• Aydogdu, M., Axial vibration of the nanorods with the nonlocal continuum rod model. Physica E: Low-dimensional Systems and Nanostructures, 41, 861-864, 2009.
• Adhikari, S., Murmu, T., McCarthy, M.A., Dynamic finite element analysis of axially vibrating nonlocal rods. Finite Elements in Analysis and Design, 630, 42-50, 2013.
• Demir, Ç., Civalek, Ö., Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Applied Mathematical Modelling, 37, 9355-9367, 2013.
• Lim, C.W., Islam, M.Z., Zhang, G., A nonlocal finite element method for torsional statics and dynamics of circular nanostructures. International Journal of Mechanical Sciences, 94-95, 232-243, 2015.
• Zhu, X., Li, L., On longitudinal dynamics of nanorods. International Journal of Engineering Science, 120, 129-145, 2017.
• Numanoğlu, H.M., Akgöz, B., Civalek, Ö., On dynamic analysis of nanorods. International Journal of Engineering Science, 130, 33-50, 2018.
• Karlicic, D.Z., Ayed, S., Flaieh, E., Nonlocal axial vibration of the multiple Bishop nanorod system. Mathematics and Mechanics of Solids, 24, 1668-1691, 2018.
• Nazemnezhad, R., Kamali, K., Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory. Steel and Composite Structures, 28, 749-758, 2018.
• Bao, S., Cao, J., Wang, S., Vibration analysis of nanorods by the Rayleigh-Ritz method and truncated Fourier series. Results in Physics, 12, 327-334, 2019.
• Numanoğlu, H.M., Civalek Ö., On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM, International Journal of Mechanical Sciences, 161-162, 105076, 2019.
• Civalek, Ö., Numanoğlu, H.M., Nonlocal finite element analysis for axial vibration of embedded Love–Bishop nanorods. International Journal of Mechanical Sciences, 188, 105939, 2020.
• Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, 288-307, 2007.
• Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N., Boumia, L., The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Journal of Physics D: Applied Physics, 41, 225404, 2008.
• Thai, H.T., A nonlocal beam theory for bending, buckling and vibration of nanobeams. International Journal of Engineering Science, 52, 56-64, 2012.
• Askari, H., Esmailzadeh, E., Zhang, D., Nonlinear vibration analysis of nonlocal nanowires. Composites Part B: Engineering, 67, 607-613, 2014.
• Mercan, K., Numanoğlu, H.M., Akgöz, B., Demir, C., Civalek, Ö., Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix. Archive of Applied Mechanics, 87, 1797-1814, 2017.
• Eltaher, M.A., Alshorbagy, A.E., Mahmoud, F.F., Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 37, 4787-4797, 2013.
• Eptaimeros, K.G., Koutsoumaris, C.C., Tsamasphyros, G.J., Nonlocal integral approach to the dynamical response of nanobeams. International Journal of Mechanical Sciences, 115-116, 68-80, 2016.
• Civalek, Ö., 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.
• Numanoğlu, H.M., Uzun, B., Civalek, Ö., Derivation of nonlocal finite element formulation for nano beams. International Journal of Engineering and Applied Sciences, 10, 131-139, 2018.
• Numanoğlu, H.M., Mercan, K, Civalek, Ö., Finite element model and size-dependent stability analysis of boron nitride and silicon carbide nanowires/nanotubes. Scientia Iranica Transactions A: Civil Engineering, 26, 2079-2099, 2019.
• Numanoğlu, H.M., Thermal vibration of zinc oxide nanowires by using nonlocal finite element method. International Journal of Engineering and Applied Sciences, 12, 99-110, 2020.
• Numanoğlu, H.M., Ersoy, H., Civalek, O., Ferreira, A.J.M., Derivation of nonlocal FEM formulation for thermo-elastic Timoshenko beams on elastic matrix. Composite Structures, 273, 114292, 2021.
• Ersoy, H, Numanoğlu, H.M., Akgöz, B., Civalek Ö., A new eigenvalue problem solver for thermo‐mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method. Mathematical Methods in the Applied Sciences, 2021.
• de Sciarra, F.M., Finite element modelling of nonlocal beams. Physica E: Low-Dimesional Systems and Nanostructures, 59, 144-149, 2014.
• Numanoğlu, H.M., Civalek Ö., On the dynamics of small-sized structures, International Journal of Engineering Science, 145, 103164, 2019.
• Hozhabrossadati, S.M., Challamel, N., Rezaiee-Pajand, M., Sani, A.A., Free vibration of a nanogrid based on Eringen’s stress gradient model. Mechanics Based Design of Structures and Machines, 2020.
• Russillo, A.F., Failla, G., Alotta, G., de Sciarra, F.M., Barretta, R., On the dynamics of nano-frames. International Journal of Engineering Science, 160, 103433, 2021.
• Numanoğlu, H.M., Mercan, K., Civalek, Ö., Frequency and mode shapes of Au nanowires using the continuous beam models. International Journal of Engineering and Applied Sciences, 9, 55-61, 2017.
• Numanoğlu, H.M., Nanoyapıların Kiriş ve Çubuk Modellerinin Yerel Olmayan Elastisite Teorisi Kullanılarak Titreşim Analizi. BSc. Thesis, Akdeniz University, Antalya, 2017.
• Numanoğlu, H.M., Civalek, Ö., Elastic beam model and bending analysis of silver nanowires. International Journal of Engineering and Applied Sciences, 10, 13-20, 2018.
• Numanoğlu, H.M., Ersoy, H, Akgöz, B., Civalek Ö., Small size and rotary inertia effects on the natural frequencies of carbon nanotubes. Curved and Layered Structures, 5, 273-279, 2018.
• Akgöz, B., Civalek Ö., Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment. Composites Part B: Engineering, 150, 68-77, 2018.
• Akgöz, B., Civalek, Ö., Investigation of size effects on static response of single-walled carbon nanotubes based on strain gradient elasticity. International Journal of Computational Methods, 9, 1240032, 2012.
• Civalek, Ö., Kiracioglu, O., Free vibration analysis of Timoshenko beams by DSC method. International Journal for Numerical Methods in Biomedical Engineering, 26, 1890-1898, 2010.
• Mercan, K., Demir, Ç., Civalek, Ö., Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique. Curved and Layered Structures, 3, 82-90, 2016.
• Dastjerdi, S., Akgöz, B., Civalek, Ö., On the effect of viscoelasticity on behavior of gyroscopes. International Journal of Engineering Science, 149, 103236, 2020.
• Civalek, Ö., Dastjerdi, S., Akbaş, Ş.D., Akgöz, B., Vibration analysis of carbon nanotube-reinforced composite microbeams. Mathematical Methods in the Applied Sciences, 2021.
• Dastjerdi, S., Akgöz, B., New static and dynamic analyses of macro and nano FGM plates using exact three-dimensional elasticity in thermal environment. Composite Structures, 192, 626-641, 2018.
• Uzun, B, Numanoğlu, H.M., Civalek, Ö., Definition of length-scale parameter in Eringen’s nonlocal elasticity via nolocal lattice and finite element formulation. International Journal of Engineering and Applied Sciences, 10, 264-275, 2018.
• Numanoğlu, H.M., Dynamic analysis of nano continuous and discrete structures based on nonlocal finite element formulation (NL-FEM). MSc. Thesis, Akdeniz University, Antalya, 2019, (In Turkish).