Karbon nanotüp takviyeli nanokirişlerin burkulma davranışının yerel olmayan elastisite ve elastik zemin etkileriyle incelenmesi

Yıl 2025, Cilt: 15 Sayı: 1, 105 - 121, 15.03.2025

Uğur Kafkas , Gökhan Güçlü

Öz

Bu çalışmada, elastik bir zemin üzerinde yer alan karbon nanotüp (KNT) takviyeli nanokirişlerin burkulma davranışı, Euler-Bernoulli kiriş teorisi çerçevesinde yerel olmayan elastisite teorisi ve yerel olmayan sonlu elemanlar yöntemi kullanılarak incelenmiştir. KNT hacim oranı, yerel olmayan parametre, elastik zemin parametresi ve uzunluk/kesit kalınlığı oranı gibi parametrelerin kritik burkulma yükü üzerindeki etkileri analiz edilmiştir. Kısa ve uzun KNT takviyeleri dikkate alınarak KNT takviyeli nanokiriş modellenmiş ve mekanik özellikler karışım kuramı kullanılarak belirlenmiştir. KNT takviyeli nanokirişlerin kritik burkulma yüklerinin belirlenebilmesi amacıyla, Euler-Bernoulli kiriş teorisine dayalı olarak yerel olmayan etkileri de içeren rijitlik matrisleri ve kuvvet vektörleri türetilmiş ve analizler bu doğrultuda gerçekleştirilmiştir. Sonuçlar, KNT hacim oranının artmasının kritik burkulma yükünü önemli ölçüde artırdığını göstermektedir, bu da KNT'lerin nanokirişleri güçlendirmede önemli bir rol oynadığını ortaya koymaktadır. Diğer yandan, yerel olmayan parametre kritik burkulma yükünü olumsuz etkilemekte ve nanokirişin burkulma dayanımını azaltmaktadır. Ancak yerel olmayan parametrenin burkulma dayanımına olan etkisi ihmal edilebilir düzeydedir. Elastik zemin parametresi ise kritik burkulma yükünü pozitif yönde etkilemekte olup, burkulma direncini artırmaktadır. Bu bulgu, elastik zeminin nanokirişlerin yapısal stabilitesini geliştirmede önemli bir rol oynadığını göstermektedir. Uzunluk/kesit kalınlığı oranı da bir diğer önemli parametre olup, uzun ve ince nanokirişlerin burkulmaya daha yatkın olduğunu ve bu oranın artmasıyla birlikte kritik burkulma yükünün azaldığını göstermektedir. Yukarıda belirtilen dört parametrenin burkulma dayanımına olan etkisi sonlu elemanlar yöntemi kullanılarak belirlendiği için bundan sonraki nano boyutlu kirişlerin sayısal olarak modellenmesi çalışmalarında elde edilen sonuçlar yol gösterici olacaktır.

Anahtar Kelimeler

Karbon nanotüp (KNT) takviyeli nanokirişler, Kritik burkulma yükü, Elastik zemin parametresi, Sonlu elemanlar yöntemi, Yerel olmayan elastisite teorisi

Investigation of buckling behavior of carbon nanotube reinforced nanobeams according to nonlocal elasticity and elastic foundation effects

Öz

In this study, the buckling behavior of carbon nanotube (CNT) reinforced nanobeams on an elastic foundation is investigated using the nonlocal elasticity theory and nonlocal finite element method within the framework of Euler-Bernoulli beam theory. The effects of CNT volume fraction, nonlocal parameter, elastic foundation parameter and length-to-thickness ratio on the critical buckling load are analyzed. CNT reinforced nanobeam is modeled considering short and long CNT reinforcements, and mechanical properties are determined by the rule of mixtures. To determine the critical buckling loads of CNT reinforced nanobeams, stiffness matrices and force vectors, including nonlocal effects, are derived based on the Euler-Bernoulli beam theory and analyses are carried out accordingly. The results show that increasing the CNT volume fraction significantly increases the critical buckling load, suggesting that CNTs play an essential role in strengthening the nanobeams. On the other hand, the nonlocal parameter negatively affects the critical buckling load and decreases the buckling strength of the nanobeam. However, the effect of the nonlocal parameter on the buckling strength is negligible. The elastic foundation parameter positively affects the critical buckling load and increases the buckling resistance. This finding indicates that the elastic foundation plays an important role in improving the structural stability of the nanobeams. The length-to-thickness ratio is another important parameter, indicating that long and thin nanobeams are more prone to buckling and the critical buckling load decreases with increasing this ratio. Since the effects of the above-mentioned four parameters on the buckling strength were determined using the finite element method, the results obtained will guide the subsequent numerical modeling studies of nano-sized beams.

Anahtar Kelimeler

Carbon nanotube (CNT) reinforced nanobeams, Critical buckling load, Elastic foundation parameter, Finite element method, Nonlocal elasticity theory

Kaynakça

• Aifantis, E. C. (1999). Strain gradient interpretation of size effects. International Journal of Fracture, 95(1–4), 299–314. https://doi.org/10.1007/978-94-011-4659-3_16
• Aria, A. I., & Friswell, M. I. (2019). A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Composites Part B: Engineering, 166, 233–246. https://doi.org/10.1016/j.compositesb.2018.11.071
• Arshid, E., Arshid, H., Amir, S. & Mousavi, S. B. (2021). Free vibration and buckling analyses of FG porous sandwich curved microbeams in thermal environment under magnetic field based on modified couple stress theory. Archives of Civil and Mechanical Engineering, 21(1), 6. https://doi:10.1007/s43452-020-00150-x
• Ashrafi, B., Hubert, P. & Vengallatore, S. (2006). Carbon nanotube-reinforced composites as structural materials for microactuators in microelectromechanical systems. Nanotechnology, 17(19), 4895-4903. https://doi:10.1088/0957-4484/17/19/019
• Balasubramanian, K. & Burghard, M. (2005). Chemically functionalized carbon nanotubes. Small, 1 2(2), 180-92. https://doi:10.1002/SMLL.200400118
• Baughman, R. H., Zakhidov, A. A. & De Heer, W. A. (2002). Carbon Nanotubes--the Route Toward Applications. Science, 297(5582), 787-792. https://doi:10.1126/SCIENCE.1060928
• Bentrar, H., Chorfi, S. M., Belalia, S. A., Tounsi, A., Ghazwani, M. H., & Alnujaie, A. (2023). Effect of porosity distribution on free vibration of functionally graded sandwich plate using the P-version of the finite element method. Structural Engineering and Mechanics, 88(6), 551–567. https://doi.org/10.12989/sem.2023.88.6.551
• Belarbi, M.-O., Houari, M.-S.-A., Daikh, A. A., Garg, A., Merzouki, T., Chalak, H. D., & Hirane, H. (2021). Nonlocal finite element model for the bending and buckling analysis of functionally graded nanobeams using a novel shear deformation theory. Composite Structures, 264, 113712. https://doi.org/10.1016/j.compstruct.2021.113712
• Casafont, M., Marimon, F., & Pastor, M. M. (2009). Calculation of pure distortional elastic buckling loads of members subjected to compression via the finite element method. Thin-Walled Structures, 47(6–7), 701–729. https://doi.org/10.1016/j.tws.2008.12.001
• Chi Tho, N., van Thom, D., Hong Cong, P., Zenkour, A. M., Hong Doan, D., & van Minh, P. (2023). Finite element modeling of the bending and vibration behavior of three-layer composite plates with a crack in the core layer. Composite Structures, 305, 116529. https://doi.org/10.1016/j.compstruct.2022.116529
• Civalek, Ö., & Numanoğlu, H. M. (2020). Nonlocal finite element analysis for axial vibration of embedded love–bishop nanorods. International Journal of Mechanical Sciences, 188, 105939. https://doi.org/10.1016/j.ijmecsci.2020.105939
• Civalek, Ö., Uzun, B., Yaylı, M. Ö., & Akgöz, B. (2020a). 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. https://doi.org/10.1140/epjp/s13360-020-00385-w
• Civalek, Ö., Uzun, B. & Yaylı, M. Ö. (2020b). Stability analysis of nanobeams placed in electromagnetic field using a finite element method. Arabian Journal of Geosciences, 13(21), 1165. https://doi:10.1007/s12517-020-06188-8
• Dai, K. Y., & Liu, G. R. (2007). Free and forced vibration analysis using the smoothed finite element method (SFEM). Journal of Sound and Vibration, 301(3–5), 803–820. https://doi.org/10.1016/j.jsv.2006.10.035
• Demir, C., Mercan, K., Numanoglu, H. M., & Civalek, O. (2018). Bending Response of Nanobeams Resting on Elastic Foundation. Journal of Applied and Computational Mechanics, 4(2), 105–114. https://doi.org/10.22055/jacm.2017.22594.1137
• Eltaher, M. A., Alshorbagy, A. E., & Mahmoud, F. F. (2013). Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 37(7), 4787–4797. https://doi.org/10.1016/j.apm.2012.10.016
• Eringen, A. C. (1967). Theory of micropolar plates. Zeitschrift Für Angewandte Mathematik Und Physik ZAMP 1967 18:1, 18(1), 12–30. https://doi.org/10.1007/BF01593891
• Esawi, A. M. K. & El Borady, M. A. (2008). Carbon nanotube-reinforced aluminium strips. Composites Science and Technology, 68(2), 486-492. https://doi:10.1016/J.COMPSCITECH.2007.06.030
• Fattahi, A. M. & Safaei, B. (2017). Buckling analysis of CNT-reinforced beams with arbitrary boundary conditions. Microsystem Technologies, 23(10), 5079-5091. https://doi:10.1007/s00542-017-3345-5
• Feng, Y., Kowalsky, M. J., & Nau, J. M. (2015). Finite-Element Method to Predict Reinforcing Bar Buckling in RC Structures. Journal of Structural Engineering, 141(5). https://doi.org/10.1061/(ASCE)ST.1943-541X.0001048
• Gurtin, M. E., Weissmüller, J., & Larché, F. (1998). A general theory of curved deformable interfaces in solids at equilibrium. Philosophical Magazine A, 78(5), 1093–1109. https://doi.org/10.1080/01418619808239977
• Kafkas, U. (2024). Değiştirilmiş Gerilme Çifti Teorisi ile Gözenekli Fonksiyonel Derecelendirilmiş Konsol Nanokirişlerin Statik Analizi. Uludağ University Journal of The Faculty of Engineering, 29(2), 393–412. https://doi.org/10.17482/uumfd.1459934
• Kahya, V., & Turan, M. (2017). Bending of Laminated Composite Beams by a Multi-Layer Finite Element Based on a Higher-Order Theory. Acta Physica Polonica A, 132(3), 473–475. https://doi.org/10.12693/APhysPolA.132.473
• Karamanli, A., & Vo, T. P. (2022). Finite element model for free vibration analysis of curved zigzag nanobeams. Composite Structures, 282, 115097. https://doi.org/10.1016/J.COMPSTRUCT.2021.115097
• Lal, A. & Markad, K. (2019). Thermo-Mechanical Post Buckling Analysis of Multiwall Carbon Nanotube-Reinforced Composite Laminated Beam under Elastic Foundation. Curved and Layered Structures, 6(1), 212-228. https://doi:10.1515/CLS-2019-0018
• Lam, D. C. C., Yang, F., Chong, A. C. M., Wang, J., & Tong, P. (2003). Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8), 1477–1508. https://doi.org/10.1016/S0022-5096(03)00053-X
• Lim, C. W., Zhang, G., & Reddy, J. N. (2015). A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298–313. https://doi.org/10.1016/J.JMPS.2015.02.001
• Mesbah, A., Belabed, Z., Amara, K., Tounsi, A., Bousahla, A. A., & Bourada, F. (2023). Formulation and evaluation a finite element model for free vibration and buckling behaviours of functionally graded porous (FGP) beams. Structural Engineering and Mechanics, 86(3), 291–309. https://doi.org/10.12989/sem.2023.86.3.291
• Mindlin, R. D. (1965). Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1(4), 417–438. https://doi.org/10.1016/0020-7683(65)90006-5
• Motevalli, B., Montazeri, A., Tavakoli-Darestani, R. & Rafii-Tabar, H. (2012). Modeling the buckling behavior of carbon nanotubes under simultaneous combination of compressive and torsional loads. Physica E-low-dimensional Systems & Nanostructures, 46, 139-148. https://doi:10.1016/J.PHYSE.2012.09.006
• Namilae, S. & Chandra, N. (2006). Role of atomic scale interfaces in the compressive behavior of carbon nanotubes in composites. Composites Science and Technology, 66(13), 2030-2038. https://doi:10.1016/j.compscitech.2006.01.009
• Phadikar, J. K. & Pradhan, S. C. (2010). Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates. Computational Materials Science, 49(3), 492-499. https://doi:10.1016/j.commatsci.2010.05.040
• Pham, Q.-H., Tran, V. K., & Nguyen, P.-C. (2022). Hygro-thermal vibration of bidirectional functionally graded porous curved beams on variable elastic foundation using generalized finite element method. Case Studies in Thermal Engineering, 40, 102478. https://doi.org/10.1016/j.csite.2022.102478
• Popov, V. N., Van Doren, V. E. & Balkanski, M. (2000). Elastic properties of crystals of single-walled carbon nanotubes. Solid State Communications, 114(7), 395-399. https://doi:10.1016/S0038-1098(00)00070-3
• Pouresmaeeli, S. & Fazelzadeh, S. A. (2017). Uncertain Buckling and Sensitivity Analysis of Functionally Graded Carbon Nanotube-Reinforced Composite Beam. International Journal of Applied Mechanics, 09(05), 1750071. https://doi:10.1142/S1758825117500715
• Pradhan, S. C. & Reddy, G. K. (2011). Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM. Computational Materials Science, 50(3), 1052-1056. https://doi:10.1016/J.COMMATSCI.2010.11.001
• Reddy, J. N. (2002). Energy Principles and Variational Methods in Applied Mechanics (2nd ed.). John Wiley and Sons.
• Reddy, J. N., Nampally, P., & Srinivasa, A. R. (2020). Nonlinear analysis of functionally graded beams using the dual mesh finite domain method and the finite element method. International Journal of Non-Linear Mechanics, 127, 103575. https://doi.org/10.1016/j.ijnonlinmec.2020.103575
• Setoodeh, A. R., Derahaki, M. & Bavi, N. (2015). DQ thermal buckling analysis of embedded curved carbon nanotubes based on nonlocal elasticity theory. Latin American Journal of Solids and Structures, 12(10), 1901-1917. https://doi:10.1590/1679-78251894
• Shen, H.-S. (2009). Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Composite Structures, 91(1), 9-19. https://doi:10.1016/j.compstruct.2009.04.026
• Song, W., Deng, Z., Wu, H., & Zhan, Y. (2022). Extended finite element modeling of hot mix asphalt based on the semi-circular bending test. Construction and Building Materials, 340, 127462. https://doi.org/10.1016/j.conbuildmat.2022.127462
• Suhr, J., Koratkar, N., Keblinski, P. & Ajayan, P. (2005). Viscoelasticity in carbon nanotube composites. Nature Materials, 4(2), 134-137. https://doi:10.1038/nmat1293
• Taghizadeh, M., Ovesy, H. R. & Ghannadpour, S. A. M. (2016). Beam Buckling Analysis by Nonlocal Integral Elasticity Finite Element Method. International Journal of Structural Stability and Dynamics, 16(06), 1550015. https://doi:10.1142/S0219455415500157
• Thang, P. T. (2019). Geometrically nonlinear buckling analysis of functionally graded carbon nanotube reinforced cylindrical panels resting on Winkler–Pasternak elastic foundation. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(2), 702-712. https://doi:10.1177/0954406218760957
• Toupin, R. (1962). Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis, 11(1), 385–414. https://doi.org/10.1007/BF00253945
• Tuna, M., & Kirca, M. (2017). Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method. Composite Structures, 179, 269–284. https://doi.org/10.1016/j.compstruct.2017.07.019
• Turan, M., & Hacıoğlu, M. İ. (2022). Buckling Analysis of Functionally Graded Beams Using the Finite Element Method. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 15(Special Issue I), 98–109. https://doi.org/10.18185/erzifbed.1199454
• Turan, M., & Hacıoğlu, M. İ. (2023). Yüksek mertebe sonlu eleman modeliyle fonksiyonel derecelendirilmiş kirişlerin serbest titreşim ve statik analizi. Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi. https://doi.org/10.17714/gumusfenbil.1185301
• Turan, M., & Kahya, V. (2018). Fonksiyonel Derecelendirilmiş Kirişlerin Serbest Titreşim Analizi. Karadeniz Fen Bilimleri Dergisi, 8(2), 119–130. https://doi.org/10.31466/kfbd.453833
• Turan, M., & Kahya, V. (2021). Fonksiyonel derecelendirilmiş sandviç kirişlerin Navier yöntemiyle serbest titreşim ve burkulma analizi. Gazi Üniversitesi Mühendislik Mimarlık Fakültesi Dergisi, 36(2), 743–758. https://doi.org/10.17341/gazimmfd.599928
• Uhm, T., & Youn, S. (2009). T‐spline finite element method for the analysis of shell structures. International Journal for Numerical Methods in Engineering, 80(4), 507–536. https://doi.org/10.1002/nme.2648
• Uzun, B., Kafkas, U., & Yaylı, M. Ö. (2021). Free vibration analysis of nanotube based sensors including rotary inertia based on the Rayleigh beam and modified couple stress theories. Microsystem Technologies, 27(5), 1913–1923. https://doi.org/10.1007/s00542-020-04961-z
• Uzun, B., Numanoglu, H., & Civalek, O. (2018). Free vibration analysis of BNNT with different cross-Sections via nonlocal FEM. Journal of Computational Applied Mechanics, 49(2), 252–260. https://doi.org/10.22059/jcamech.2018.266789.328
• Uzun, B., & Yaylı, M. Ö. (2022). A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory. International Journal of Engineering and Applied Sciences, 14(1), 1–14. https://doi.org/10.24107/ijeas.1064690
• Uzun, B., Yaylı, M. Ö., & Deliktaş, B. (2020). Free vibration of FG nanobeam using a finite‐element method. Micro & Nano Letters, 15(1), 35–40. https://doi.org/10.1049/mnl.2019.0273
• Wang, Q., Varadan, V. K., & Quek, S. T. (2006). Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Physics Letters A, 357(2), 130–135. https://doi.org/10.1016/j.physleta.2006.04.026
• Wang, Q., Liew, K. M., He, X. Q. & Xiang, Y. (2007). Local buckling of carbon nanotubes under bending. Applied Physics Letters, 91(9). https://doi:10.1063/1.2778546
• Wattanasakulpong, N. & Ungbhakorn, V. (2013). Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Computational Materials Science, 71, 201-208. https://doi:10.1016/J.COMMATSCI.2013.01.028
• Wu, C.-P. & Yu, J.-J. (2019). A review of mechanical analyses of rectangular nanobeams and single-, double-, and multi-walled carbon nanotubes using Eringen’s nonlocal elasticity theory. Archive of Applied Mechanics, 89(9), 1761-1792. https://doi:10.1007/s00419-019-01542-z
• Yang, F., Chong, A. C. M., Lam, D. C. C., & Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39(10), 2731–2743. https://doi.org/10.1016/S0020-7683(02)00152-X
• Yaylacı, M., Yaylacı, E. U., Özdemir, M. E., Öztürk, Ş., & Sesli, H. (2023). Vibration and buckling analyses of FGM beam with edge crack: Finite element and multilayer perceptron methods. Steel and Composite Structures, 46(4), 565–575. https://doi.org/10.12989/scs.2023.46.4.565
• Yaylı, M.Ö. (2015). Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube. Acta Physica Polonica A, 127(3), 678-683. https://doi:10.12693/APhysPolA.127.678
• Yaylı, M. Ö. (2017). Buckling analysis of a cantilever single‐walled carbon nanotube embedded in an elastic medium with an attached spring. Micro & Nano Letters, 12(4), 255-259. https://doi:10.1049/mnl.2016.0662
• Yaylı, M. Ö. (2019). Stability analysis of a rotationally restrained microbar embedded in an elastic matrix using strain gradient elasticity. Curved and Layered Structures, 6(1), 1-10. https://doi:10.1515/cls-2019-0001
• Zhu, S., Dong, R., Liu, Z., Liu, H., Lu, Z., & Guo, Y. (2024). A finite element method study of the effect of vibration on the dynamic biomechanical response of the lumbar spine. Clinical Biomechanics, 111, 106164. https://doi.org/10.1016/j.clinbiomech.2023.106164