AĞIRLIKLI ARTIKLAR KULLANILARAK NANOÇUBUKLARIN EKSENEL STATİK ANALİZİ İÇİN KESİN ÇÖZÜMLER

Yıl 2021, , 588 - 598, 20.06.2021

Mustafa Özgür Yaylı , Uğur Kafkas , Büşra Uzun

https://doi.org/10.21923/jesd.719059

Öz

Bu çalışmada, Eringen’in yerel olmayan diferansiyel modeli kullanılarak; üçhen yayılı yüklenmiş nano çubukların eksenel statik analizi verilmiştir. Üç ağırlıklı artık tabanlı yöntem (Subdomain, Galerkin ve Least squares yöntemleri) gerçek statik deplasmanı elde etmek için kullanılmıştır. Bu yöntemler bölgenin tamamında integral hatalarını minimize etme varsayımına dayanmaktadır. Sistem denklemleri çözümü aranan bilinmeyenler ile aynı sayıda olmadır. Bu yüzden üç ağırlıklı artık yöntemi için de kübik polinomlar statik deplasmanı göstermek üzere seçilmiştir. Subdomain, Galerkin and Least squares yöntemleriyle gerçek çözümler ile aynı polinomlar olarak elde edilmiştir. Değişik sayıda bilinmeyen içeren sabitler ile grafikler çizdirilerek çözümler gösterilmiştir.Bu çalışmada, Eringen’in yerel olmayan diferansiyel modeli kullanılarak; üçhen yayılı yüklenmiş nano çubukların eksenel statik analizi verilmiştir. Üç ağırlıklı artık tabanlı yöntem (Subdomain, Galerkin ve Least squares yöntemleri) gerçek statik deplasmanı elde etmek için kullanılmıştır. Bu yöntemler bölgenin tamamında integral hatalarını minimize etme varsayımına dayanmaktadır. Sistem denklemleri çözümü aranan bilinmeyenler ile aynı sayıda olmadır. Bu yüzden üç ağırlıklı artık yöntemi için de kübik polinomlar statik deplasmanı göstermek üzere seçilmiştir. Subdomain, Galerkin and Least squares yöntemleriyle gerçek çözümler ile aynı polinomlar olarak elde edilmiştir. Değişik sayıda bilinmeyen içeren sabitler ile grafikler çizdirilerek çözümler gösterilmiştir.

Anahtar Kelimeler

Eksenel Statik, Eksenel Statik, Nanoçubuk, Ağırlıklı Artık Yöntemleri, Yer Değiştirme Fonksiyonu, Gerçek Çözüm

EXACT SOLUTIONS FOR AXIAL STATIC ANALYSIS OF NANORODS USING WEIGHTED RESIDUALS

Öz

In the present work, axial static analysis of nanorods under triangular loading is presented via Eringen’s nonlocal differential model. Three weighted residual methods (Subdomain, Galerkin and Least squares methods) are used to obtain the exact static deflection. These methods require that the integral of the error with different assumptions over the domain be set to zero. The number of equations have to be equal to unknown terms. A cubic displacement function has been chosen for three weighted residual methods. Subdomain, Galerkin and Least squares methods yield identical solution as the exact solution. The plots of the solution are shown for different number of unknown coefficients.

Anahtar Kelimeler

Axial Static, Nanorod, Weighted Residual Methods, Displacement Function, Exact Solution

Kaynakça

• Akbaş, Ş. D. 2019. Longitudinal forced vibration analysis of porous a nanorod Mühendislik Bilimleri ve Tasarım Dergisi, 7(4), 736-743.
• Akgöz, B. 2019. Ritz yöntemi ile değişken kesitli kolonların burkulma analizi Mühendislik Bilimleri ve Tasarım Dergisi, 7(2), 452-458.
• Akgöz, B., Civalek, Ö., 2013. Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity. Structural Engineering and Mechanics, 48, 195–205.
• Akgöz, B., Civalek, Ö., 2015. Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Composite Structures, 134, 294–301.
• Ansari, R., Gholami, R., Sahmani, S., 2013. Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory. Arch Appl Mech, 83, 1439–1449.
• Arda, M., Aydogdu., M, 2016. Bending of CNTs Under The Partial Uniform Load. Int J Eng Appl Sci, 8, 21–21.
• Arda, M., Aydogdu. M., 2017. Longitudinal Vibration of CNTs Viscously Damped in Span. Int J Eng Appl Sci, 9, 22–22.
• Arefi, M., Firouzeh, S., Bidgoli, E.M.R., Civalek, Ö. 2020. Analysis of Porous Micro-plates Reinforced with FG-GNPs Based on Reddy plate Theory. Composite Structures, 112391.
• Aydoğdu, M., 2009. Axial vibration of the nanorods with the nonlocal continuum rod model. Physica-E Low-dimensional Systems and Nanostructures, 41, 861–864.
• Aydogdu, M., Arda, M., 2014. Torsional statics and dynamics of nanotubes embedded in an elastic medium. Compos Struct, 114, 80–91.
• Aydogdu, M., Arda, M., 2016. Forced vibration of nanorods using nonlocal elasticity. Adv nano Res, 4, 265–279.
• Aydogdu, M., Arda, M., Filiz, S., 2018. Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter. Adv Nano Res, 6, 257–278.
• Civalek, Ö., Demir, Ç., 2016. A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation, 289, 335–352.
• Civalek, Ö., Uzun, B., Yaylı, M.Ö., Akgöz, B. 2020. Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method. The European Physical Journal Plus, 135(4), 381.
• Dastjerdi, S., Akgöz, B., Civalek, Ö. 2020. On the effect of viscoelasticity on behavior of gyroscopes. International Journal of Engineering Science, 149, 103236.
• Demir, C., Civalek, Ö. 2013. Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models. Applied Mathematical Modelling, 37, 9355-9367.
• Ebrahimi, F., Barati, M.R., Civalek, Ö. 2019. Application of Chebyshev–Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures. Engineering with Computers, 1-12.
• Ece, M.C., Aydogdu, M., 2007. Nonlocal elasticity effect on vibration of in-plane loaded double- walled carbon nano-tubes. Acta Mech., 190, 185–195.
• Eringen, A.C., Edelen, D.G.B., 1972. On nonlocal elasticity. International Journal of Engineering Science, 10, 233–248.
• Eringen, A.C., 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface-waves. J. Appl. Phys., 54, 4703–4710.
• Jalaei, M.H., Civalek, Ö., 2019. A nonlocal strain gradient refined plate theory for dynamic instability of embedded graphene sheet including thermal effects. Composite Structures, 220, 209-220.
• Kounadis, A.N., Mallis, J., Sbarounis, A., 2006. Postbuckling analysis of columns resting on an elastic foundation. Arch Appl Mech, 75, 395–404.
• Li, C., Yao, L.Q., Chen, W.Q., Li, S., 2015. Comments on nonlocal effects in nano-cantilever beams. International Journal of Engineering Science, 87, 47–57.
• Li, C. 2014. A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries. Composite Structures, 118, 607–621.
• Li, C., Liu, J.J., Cheng, M., Fan, X.L., 2017. Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces. Composites Part B-Engineering, 116, 153–169.
• Liu, J.J., Li, C., Fan, X.L., Tong, L.H., 2017. Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory. Applied Mathematics and Computation, 45, 65–84.
• Reddy, J.N., Pang, S.D., 2008. Nonlocal continuum theories of beam for the analysis of carbon nanotubes. Journal of Applied Physics, 103, 1–16.
• Şimşek, M., 2007. Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory. Physica-E Low-dimensional Systems and Nanostructures, 43, 182–191.
• Thai, S., Thai, H.T., Vo, T.P., Lee, S., 2018. Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis. Composite Structures, 201, 13-20.
• Uzun, B., Yaylı, M.Ö., 2020a. A solution method for longitudinal vibrations of functionally graded nanorods. International Journal of Engineering and Applied Sciences, 12, 78-87.
• Uzun, B., Yaylı, M.Ö. 2020b. Nonlocal vibration analysis of Ti-6Al-4V/ZrO2 functionally graded nanobeam on elastic matrix. Arabian Journal of Geosciences, 13(4), 1-10.
• Uzun, B., Kafkas, U., Yaylı, M.Ö. 2020a. Axial dynamic analysis of a Bishop nanorod with arbitrary boundary conditions. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, e202000039.
• Uzun, B., Yaylı, M.Ö., Deliktaş, B. 2020b. Free vibration of FG nanobeam using a finite-element method. Micro & Nano Letters, 15(1), 35-40.
• Uzun, B., Civalek, Ö., Yaylı, M.Ö. 2020c. Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions. Mechanics Based Design of Structures and Machines, 1-20.
• Uzun, B., Kafkas, U., Yaylı, M.Ö. 2020d. Free vibration analysis of nanotube based sensors including rotary inertia based on the Rayleigh beam and modified couple stress theories. Microsystem Technologies, 1-11.
• Uzun, B., Kafkas, U., Yaylı, M.Ö. 2020e. Stability analysis of restrained nanotubes placed in electromagnetic field. Microsystem Technologies, 1-12.
• Wang, Q., Liew, K.M., 2007. Application of nonlocal continuum mechanics to static analysis of micro and nano-structures. Phys. Lett. A, 363, 236–242.
• Yayli, M.Ö., 2014. On the axial vibration of carbon nanotubes with different boundary conditions. IET Micro and Nano Letters, 9, 807–811.
• Yayli, M.Ö., Yanik, F., Kandemir, S.Y., 2015. Longitudinal vibration of nanorods embedded in an elastic medium with elastic restraints at both ends. IET Micro and Nano Letters, 10, 641–644.
• Yayli, M.Ö., 2016a. Buckling analysis of a microbeam embedded in an elastic medium with deformable boundary conditions. IET Micro and Nano Letters, 11, 741–745.
• Yayli, M.Ö., 2016b. A compact analytical method for vibration analysis of single-walled carbon nanotubes with restrained boundary conditions. Journal of Vibration and Control, 22, 2542–2555.
• Yayli, M.Ö., 2017. Buckling analysis of a cantilever single-walled carbon nanotube embedded in an elastic medium with an attached spring. IET Micro and Nano Letters, 12, 255–259.
• Yoon, J., Ru, C.Q., Mioduchowski, A., 2003. Vibration of an embedded multiwall carbon nanotube. Compos Sci Technol, 63, 1533–1542.