Membran model kullanılarak grafen tabakaların titreşim hesabı

Öz

Bu çalışmada grafen tabakaların membran gibi modellenerek serbest titreşim analizleri yapılmıştır. Membranlar eğilmeye ya da burkulmaya karşı rijitliği olmayan ince plaklardır. Yanal güçleri eksenel ve merkezi kesme kuvvetleri ile taşırlar. Böyle yük taşımaları, aşırı incelikleri ve moment taşıma kapasitelerinin ihmal edilebilir olmasından dolayı gergin kablo ağlarına benzetilebilirler. Grafen tabakalar dikdörtgen ve kare geometriye sahip olmak üzere değişik boyutlarda modellenmiştir. Elde edilen denklemin çözümünde hem ayrık tekil konvolüsyon yöntemi ve hem de analitik yöntem kullanılmıştır. Literatürde bulunan plak modeli ile ilk defa yapılan membran modelinin sonuçları karşılaştırılmıştır. Bulunan değerler grafik ve tablo halinde sunulmuştur.

Anahtar Kelimeler

• Membran,Grafen,Titreşim,Modal analiz,Mod şekli

Vibration analysis of graphene sheets using membrane model

Abstract

In this present study vibration analysis of graphene sheets have been carried out by modeling as membrane model. Membranes are thin plates without the stiffness against bending and buckling. They carry lateral forces with axial and central shear forces. This specification, its extreme thinness and negligible moment capacity of membranes can be likened to the tense cable network. Graphene sheets are modeled in square and rectangular geometry. The resulting equation have been solved both analytically and the method of discrete singular convolution. The firstly obtained membrane results have been compared with results obtained by plate models in the literature. Results are given in graphics and tables.

Keywords

• Membrane,Graphene,Vibration,Modal analysis,Mode shape

Kaynakça

1. Güneşoğlu C. “Nanoteknoloji ve tekstil sektöründeki uygulamalari”. Mühendis ve Makina, 50(591), 25-34, 2009.
2. Kutucu B. Nanoteknoloji ve Çift Duvarlı Karbon Nanotüplerin Incelenmesi. Yüksek Lisans Tezi, İstanbul Teknik Üniversitesi Fen Bilimleri Enstitüsü, İstanbul, Türkiye, 2010.
3. Erkoç Ş. Nanobilim ve Nanoteknoloji. Üçüncü Baskı, Ankara, Türkiye, ODTÜ Yayıncılık, 2008.
4. Moğulkoç A. Grafende Kütlesiz Dirac Fermiyonları Gazı. Yüksek Lisans Tezi, Ankara Üniversitesi Fen Bilimleri Enstitüsü, Ankara, Türkiye, 2008.
5. Wikipedia. “Graphene”. https://en.wikipedia.org/wiki/Graphene (20.12.2015).
6. Geim AK, Novoselov KS. “The rise of graphene”. Nature Materials, 6(3), 183-191, 2007.
7. Novoselov KSA, et al. "Two-dimensional gas of massless Dirac fermions in graphene". Nature, 438(7065), 197-200, 2005.
8. Portugal R, Golebiowski L, Frenkel D. “Oscillation of membranes using computer algebra”. American Journal of Physics, 67(6), 534-537, 1999.

9. Young AF, Kim P. "Quantum interference and Klein tunnelling in graphene heterojunctions". Nature Physics, 5, 222-226, 2009.
10. Zhang Y, Tan YW, Stormer HL, Kim P. "Experimental observation of the quantum Hall effect and Berry’s phase in graphene". Nature, 438, 201-204, 2005.
11. Rao SS. Vibration of Continuous Systems. 1st ed. Hokoben, New Jersey, USA, John Wiley & Sons, 2007.
12. Leissa AW, Qatu MS. Vibration of Continuous Systems. 1st ed. New York, McGraw-Hill Education, 2011.
13. Akgöz B, Civalek O. “Frequency response of skew and trapezoidal shaped mono-layer graphene sheets via discrete singular convolution”. Scientia Iranica, Transaction F- Nanotechnology, 21(3), 1197-1207, 2014.
14. Demir Ç, Civalek O. “Tek Katmanlı Grafen Tabakaların Eğilme Ve Titreşimi”. Mühendislik Bilimleri ve Tasarım Dergisi, 4(3), 173-183, 2016.
15. Akgöz B, Civalek O. “Shear deformation beam models for functionally graded microbeams with new shear correction factors”. Composite Structures, 112, 214-225, 2014.
16. Akgöz B, Civalek O. “Thermo-mechanical buckling behavior of functionally graded microbeams embedded in elastic medium”. International Journal of Engineering Science, 85, 90-104, 2014.
17. Aksencer T, Aydogdu M. “Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory”. Physica E: Low-Dimensional Systems and Nanostructures, 43(4), 954-959, 2011.
18. Civalek O, Demir Ç. “Buckling and bending analyses of cantilever carbon nanotubes using the euler-bernoulli beam theory based on non-local continuum model”. Asian Journal of Civil Engineering (Building and Housing), 12 (5), 651-661, 2011.
19. Civalek O, Demir Ç, Akgöz B. “Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory”. International Journal of Engineering and Applied Sciences, 1(2), 47-56, 2009.
20. Mercan K, Demir Ç, Akgöz B, Civalek O. “Coordinate Transformation for Sector and Annular Sector Shaped Graphene Sheets on Silicone Matrix”. International Journal of Engineering & Applied Sciences, 7(2), 56-73, 2015.
21. 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.
22. Demir Ç. Bending and Free Vibration Analysis of Nano and Micro Structures Based on Nonlocal Elasticity Theory. MSc Thesis, Graduate School of Natural and Applied Sciences, Akdeniz University, Antalya, Turkey, 2012.
23. Demir Ç, Civalek O. “Nonlocal deflection of microtubules under point load”. International Journal of Engineering & Applied Sciences, 7(3), 33-38, 2015.
24. Emsen E, Mercan K, Akgöz B, Civalek O. “Modal analysis of tapered beam-column embedded in Winkler elastic foundation”. International Journal of Engineering & Applied Sciences, 7(2), 56-73, 2015.
25. Peddieson J, Buchanan GR, McNitt RP. “Application of nonlocal continuum models to nanotechnology”. International Journal of Engineering Science, 41(3-5), 305-312, 2003.
26. Kharagpur IIT. “Module 4: Vibrations of membranes, Lecture32-The Rectangular Membrane”, Course Note, 2012.
27. Wei GW. “Discrete singular convolution for the solution of the Fokker-Planck equations”. Journal of Chemical Physics, 110(18), 8930-8942, 1999.
28. Wei GW. “Solving quantum eigenvalue problems by discrete singular convolution”. Journal of Physics B: At. Mol.Opt.Physics, 33(3), 343-352, 2000.
29. Wei GW. “Discrete singular convolution for the Sine-Gordon equation”. Physica D, 137(3-4), 247-259, 2000.
30. Wei GW. “A unifed approach for the solution of the fokker-planck equation”. Journal of Physics A: Math. Gen., 33(27), 4935-4953, 2000.
31. Wei GW. “Wavelets generated by using discrete singular convolution kernels”. Journal of Physics A: Mathematical and General, 33(47), 8577-8596, 2000.
32. Wei GW, Zhao YB, Xiang Y. “Discrete singular convolution and its application to the analysis of plates with internal supports; Part 1: Theory and algorithm”. International Journal for Numerical Methods in Engineering, 55(8), 913-946, 2002.
33. Wei GW, Zhao YB, Xiang Y. “A novel approach for the analysis of high-frequency vibrations”. Journal of Sound and Vibration, 257(2), 207-246, 2002.
34. Wei GW. “Vibration analysis by discrete singular convolution”. Journal of Sound and Vibration, 244(3), 535-553, 2001.
35. Wei GW. “Discrete singular convolution for beam analysis”. Engineering Structures, 23, 1045-1053, 2001b.
36. Civalek O, Gürses M. “Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique”. International Journal of Pressure Vessels and Piping, 86(10), 677-683, 2009.
37. Wei GW, Zhao YB, Xiang Y. “The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution”. International Journal of Mechanical Sciences, 43, 1731-1746, 2001.
38. Civalek O. “The determination of frequencies of laminated conical shells via the discrete singular convolution method”. Journal of Mechanics of Materials and Structures, 1(1), 163-182, 2006.
39. Yunshan H, Wei GW, Xiang Y. “DSC-Ritz method for the free vibration analysis of Mindlin plates”. International Journal for Numerical Methods in Engineering, 62(2), 262-288, 2005.
40. Lim CW, Li ZR, Wei GW. “DSC-Ritz method for high-mode frequency analysis of thick shallow shells”. International Journal for Numerical Methods in Engineering, 62(2), 205-232, 2005.
41. Civalek O. “Free vibration and buckling analysis of composite plates with straight-sided quadrilateral domain based on DSC approach”. Finite Elements in Analysis and Design, 43(13), 1013-1022, 2007.
42. Civalek O. “Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method”. Applied Mathematical Modelling, 33(10), 3825-3835, 2009.
43. Zhao S, Wei GW, Xiang Y. “DSC analysis of free-edged beams by an iteratively matched boundary method”. Journal of Sound Vibration, 284(1-2), 487-493, 2005.
44. Ansari R, Sahmani S, Arash B. “Nonlocal plate model for free vibrations of single-layered graphene sheets”. Physics Letters A, 375(1), 53-62, 2010.
45. Civalek O, Akgöz B. “Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix”. 77, 295-303, 2013.
46. Akgöz B, Civalek O. “Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory”. Materials & Design, 42, 164-171, 2012.
47. Shen ZB, Tang HL, Li DK, Tang GJ. “Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal kirchhoff plate theory”. Computational Materials Sciences, 61, 200-205, 2012.
48. Behfar K, Naghdabadi N. “Nanoscale vibrational analysis of a multi-layered graphene sheet embedded in an elastic medium”. Composite Sciences and Technology, 65(7-8), 1159-1164, 2005.