CC BY
51
УДК 539.3
Анализ эффектов поверхности и масштабного фактора при поперечном и продольном изгибе нанопроволок в рамках нелокальных теорий сдвиговых деформаций
A. Lounis1, D.O. Youcef1, A.A. Bousahla1, F. Bourada1,2, A. Kaci1,3, H. Heireche1, Abdeldjebbar Tounsi1, K.H. Benrahou1, Abdelouahed Tounsi1,4,5, and M. Hussain6
1 Университет Сиди-Бель-Аббеса, Сиди-Бель-Аббес, 22000, Алжир 2 Университет Тисемсильта, Тисемсильт, 38004, Алжир 3 Университет докт. Тахара Мулая, Саида, 20000, Алжир 4 Университет Ёнсе, Сеул, 03722, Южная Корея 5 Университет нефти и полезных ископаемых им. короля Фахда, Дахран, 31261, Саудовская Аравия 6 Правительственный колледж университета Фейсалабада, Фейсалабад, 38000, Пакистан
В статье представлен анализ поперечного и продольного изгиба свободно опертых нанопроволок в рамках классических и неклассических теорий сдвиговых деформаций высшего порядка. Одномерная структура моделируется с учетом эффектов поверхности на основе теории поверхностной упругости Gurtin-Murdoch (неклассическая теория балок) и масштабного фактора на основе нелокальной теории Eringen (нелокальная теория балок), при этом поперечное смещение делится на две компоненты изгиба и сдвига. На основе принципа минимума полной потенциальной энергии получены система определяющих уравнений и ее решение с помощью решений Навье. Проведено сравнение некоторых полученных результатов с литературными данными. Показано, что поверхностные эффекты оказывают большее влияние при поперечном и продольном изгибе нанопроволок по сравнению с влиянием нелокального параметра.
Ключевые слова: нелокальный эффект, поверхностный эффект, нанопроволока, расчет на устойчивость, отклик на изгиб
DOI 10.24412/1683-805X-2021-3-36-39
Surface and small scale impacts on the bending and buckling of nanowires using various nonlocal HSDTs
A. Lounis1, D.O. Youcef1, A.A. Bousahla1, F. Bourada2,3, A. Kaci2,4, H. Heireche1,
2 2 2 5 6 7
Abdeldjebbar Tounsi , K.H. Benrahou , Abdelouahed Tounsi , , , and M. Hussain
1 Laboratoire de Modélisation et Simulation Multi-échelle, Université de Sidi Bel Abbés, Sidi Bel Abbés, 22000, Algeria 2 Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes, Sidi Bel Abbés, 22000, Algeria 3 Département des Sciences et de la Technologie, Université de Tissemsilt, Ben Hamouda, 38004, Algérie 4 Faculté de Technologie, Département de Génie Civil et Hydraulique, Université Dr Tahar Moulay, Saida, 20000, Algérie 5 Yonsei Frontier Lab., Yonsei University, Seoul, 03722, Korea
6 Department of Civil and Environmental Engineering, King Fahd University of Petroleum & Minerals,
Dhahran, 31261, Saudi Arabia
7 Department of Mathematics, Government College University Faisalabad, Faisalabad, 38000, Pakistan
This paper presents the bending and buckling analyses of simply supported nanowires using various classical and non-classical higher-order shear deformation theories (HSDTs). The one dimensional structure is modeled with including the surface effects based on the Gurtin-Murdoch surface elasticity theory (non-classical beam theory) and the small scale effect based on the Eringen nonlocal theory (nonlocal beam theory), the transverse displacement is divided into two bending and shear components. A system of governing equations is derived with the help of the minimum total potential energy principle and resolved via Navier's solutions. Several numerical results are presented and compared with those given in the literature. The results showed that the influence of the surface effects on the bending and buckling load of nanowires is more than that of the nonlocal parameter.
Keywords: nonlocal effect, surface effect, nanowires, buckling analysis, flexural response
© Lounis A., Youcef D.O., Bousahla A.A., Bourada F., Kaci A., Heireche H., Tounsi A., Benrahou K.H., Tounsi A., Hussain M., 2021
1. Introduction
Nanowires (NWs) belong to the family of thin materials that have a dimension 1D such as nano-beams and nanotubes (NT). A wide range of uses of nanowire-based devices (NWs) exists in the fields of engineering, physics applications and others sectors. Nanowires are commonly used in others devices based on advanced technological such as transistors, sensors, resonators in nano- (NEMS) and micro-electro-mechanical (MEMS) systems and actuators [1-4]. It has wide applications in environmental monitoring, medical diagnostics, food processing, mining, bioengineering and defense [5, 6].
For the first time, Gurtin and Murdoch [7, 8] proposed a generic theoretical framework based on the concept of continuum mechanics that represent the surface energy/interface. As is well known, in the structures of the nanoscale, the surface/volume ratio is high. Hence, the surface effect is one of the most important influences on the nanostructures compared to those on the macroscale. This way were clearly indicated and explained by the author Ansari and Sahmani [9] adopted various theories of beams for the analysis of the buckling of nanobeams with surface effect. and others, for example Song et al. [10] used a continuum model for the mechanical behavior of nanowires including surface and surface induced initial stresses, Dingreville et al. [11] demonstrated that the structure size influences general elastic behavior and this dependence is important when at least one of the structural dimensions contracts at nanometers.
The Nonlocal elasticity has been used widely to study the wave propagation in composites, elastic waves, dislocation mechanics and dynamic and static responses of FG-structures, carbon nanotubes, micro-tubules and nanorods, For example [12-14], as well as, Reddy and Pang [15] have modified the analytical models (EBT and TBT) using the non-local elasticity theory of Eringen's to analyze the static and dynamic behaviors of CNTs with various boundary conditions, Phadikar and Pradhan [16] analyzed the static bending and buckling of nanobeams and nano-plates in which the nonlocal elasticity is included. Civalek and Demir [17] investigated the static flexural behavior of microtubules (MTs) using the nonlocal elasticity, classical beam theory and method of differential quadrature (DQM). Based on FEM, nonlocal continuum model, Demir and Civalek [18] analyzed the longitudinal and torsional frequency and wave response of microtubules. Attia [19] developed a new analytical model to examine the responses of PFG-nanobeam based on classical EBT, modified couple
stress, nonlocal and surface elasticity. Based on EBT model and Eringen's non local elasticity, Dihaj et al. [20] investigated the vibrational response of chiral DW-CNT resting on Winkler-elastic foundation. The vibrational analysis of armchair SW-CNT in thermal environment and embedded in elastic medium is examined by Hamidi et al. [21] employing the nonlocal Timoshenko beam theory. Ebrahimi et al. [22] studied the stability and free vibrational responses of simply supported, simply-clamped and clamped-clamped FG nanobeam using HSDT model, nonlocal elasticity and Chebyshev-Ritz method. Based on nonlocal elasticity, Bensattalah et al. [23] examined analytically the effects of the small-scale coefficient, vibrational mode and geometry parameters on frequency of chiral SW-CNT using TBT formulation and nonlocal elasticity. Civalek et al. [24] examined the free vibrational response of silica carbide and carbone nanotube with various boundary conditions using EBT formulation, Hamilton's principle and finite element-method. Based on nonlocal continuum elasticity and FSDT formulation (Timoshenko's model), Bensattalah et al. [25] have examined the mechanical stability of zigzag TW-CNTs. Shanab et al. [26] studied the static (bending, buckling) and dynamic (free vibration) behaviors of FG-nanobeam on Winkler-Pasternak elastic foundation using Timo-shenko's beam theory with including the effects of surface energy and microstructure.
In this investigation, several non-classical HSDTS (shear deformation theories) are developed to examine buckling and bending responses of nonlocal nanowires (NWs) with taking into account the surface-stress and small scale effects. The accuracy on the current model is checked by comparing the obtained results with those found in the literature. The influences of the parameter of small scale and surface stress effects on the critical buckling load and maximum center deflection are examined and discussed in detail through several numerical examples.
References
1. Lee Z., Ophus C., Fischer L.M., Nelson-Fitzpatrick N., We-stra K.L., Evoy S., Radmilovic V., Dahmen U., Mitlin D. Metallic NEMS components fabricated from nanocomposite Al-Mo films // Nanotechnology. - 2006. - V. 17. -No. 12. - P. 3063-3070.
2. Witvrouw A., Mehta A. The use of functionally graded poly-SiGe layers for MEMS applications // Mater. Sci. Forum. -2005. - V. 492-493. - P. 255-260.
3. CraigheadH.G. Nanoelectromechanical systems // Science. -2000. - V. 290. - No. 5496. - P. 1532-1535.
4. Ekinci K.L., Roukes M.L. Nanoelectromechanical systems // Rev. Sci. Instrum. - 2005. - V. 76. - No. 6. - P. 061101.
5. Ahn M.-W, Park K.-S, Heo J.-H, Park J.-G, Kim D.-W, Choi K.J., Lee J.H., Hong S.-H. Gas sensing properties of defect-controlled ZnO-nanowire gas sensor // Appl. Phys. Lett. -2008. - V. 93. - No. 26. - P. 263103.
6. Venkatesan B.M., Dorvel B., Yemenicioglu S., WatkinsN., Petrov I., Bashir R. Highly sensitive, mechanically stable nanopore sensors for DNA analysis // Adv. Mater. - 2009. -V. 21. - No. 27. - P. 2771-2776.
7. Gurtin M.E., Murdoch A.I. A continuum theory of elastic material surfaces // Arch. Ration. Mech. Anal. - 1975. -V. 57. - No. 4. - P. 291-323.
8. Gurtin M.E., Murdoch A.I. Surface stress in solids // Int. J. Solids Struct. - 1978. - V. 14. - No. 6. - P. 431-440.
9. Ansari R., Sahmani S. Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories // Int. J. Eng. Sci. - 2011. - V. 49. -P. 1244-1255.
10. Song F., Huang G.L., Park H.S., Liu X.N. A continuum model for the mechanical behavior of nanowires including surface and surface induced initial stresses // Int. J. Solids Struct. - 2011. - V. 48. - P. 2154-2163.
11. Dingreville R., Qu J., Cherkaoui M. Surface free energy and its effect on the elastic behavior of nanosized particles, wires and films // J. Mech. Phys. Solids. - 2005. - V. 53. -P. 1827-1854.
12. Peddieson J., Buchanan G.R., McNitt R.P. Application of nonlocal continuum models to nanotechnology // Int. J. Eng. Sci. - 2003. - V. 41. - No. 3-5. - P. 305-312.
13. Sudak L.J. Column buckling of multiwalled carbon nano-tubes using nonlocal continuum mechanics // J. Appl. Phys. - 2003. - V. 94. - No. 11. - P. 7281-7287.
14. Aydogdu M. A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration // Physica. E. - 2009. - V. 41. - P. 1651-1655.
15. Reddy J.N., Pang S.D. Nonlocal continuum theories of beams for the analysis of carbon nanotubes // J. Appl. Phys. - 2008. - V. 103. - No. 2. - P. 023511.
16. Phadikar J.K., Pradhan S.C. Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates // Comput. Mater. Sci. - 2010. -V. 49. - No. 3. -P. 492-499.
17. Civalek O., Demir C. Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory // Appl. Math. Model. - 2011. - V. 35. - No. 5. - P. 2053-2067.
18. Demir C., Civalek O. Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models // Appl. Math. Model. - 2013. - V. 37. - No. 22. - P. 9355-9367.
19. Attia M.A. On the mechanics of functionally graded nano-beams with the account of surface elasticity // Int. J. Eng. Sci. - 2017. - V. 115. - P. 73-101.
20. Dihaj A., Zidour M., Meradjah M., Rakrak K., Heireche H., Chemi A. Free vibration analysis of chiral double-walled carbon nanotube embedded in an elastic medium using nonlocal elasticity theory and Euler-Bernoulli beam model // Struct. Eng. Mech. - 2018. - V. 65. - No. 3. - P. 335-342.
21. Hamidi A., Zidour M., Bouakkaz K., Bensattalah T. Thermal and small-scale effects on vibration of embedded armchair single-walled carbon nanotubes // J. Nano Res. - 2018. -V. 51. - P. 24-38.
22. Ebrahimi F., Barati M.R., Civalek O. Application of Cheby-shev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures // Eng.
Comput. - 2019. - https://doi.org/10.1007/s00366-019-00 742-z
23. Bensattalah T., Zidour M., Daouadji T.H. A new nonlocal beam model for free vibration analysis of chiral singlewalled carbon nanotubes // Compos. Mater. Eng. - 2019. -V. 1. - No. 1. - P. 21-31.
24. Civalek O., Uzun B., Yayli M.O., Akgöz B. Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method // Eur. Phys. J. Plus. - 2020. - V. 135. - P. 381.
25. Bensattalah T., Hamidi A., Bouakkaz K., Zidour M., Daouadji T.H. Critical buckling load of triple-walled carbon nanotube based on nonlocal elasticity theory // J. Nano Res. - 2020. - V. 62. - P. 108-119.
26. Shanab R.A., Attia M.A., Mohamed S.A., MohamedN.A. Effect of microstructure and surface energy on the static and dynamic characteristics of FG Timoshenko nanobeam embedded in an elastic medium // J. Nano Res. - 2020. -V. 61. - P. 97-117.
27. Eringen A.C. Nonlocal polar elastic continua // Int. J. Eng. Sci. - 1972. - V. 10. - P. 1-16.
28. Eringen A.C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves // J. Appl. Phys. - 1983. - V. 54. - P. 4703-4710.
29. Eringen A.C., Edelen D.G.B. On nonlocal elasticity // Int. J. Eng. Sci. - 1972. - V. 10. - P. 233-248.
30. Eringen A.C. Nonlocal Continuum Field Theories. - New York: Springer-Verlag, 2002.
31. Kar V.R., Panda S.K. Large deformation bending analysis of functionally graded spherical shell using FEM // Struct. Eng. Mech. - 2015. - V. 53. - No. 4. - P. 661-679.
32. Chandra B.M., Ramji K., Kar V.R., Panda S.K., Lalepalli K.A., Pandey H.K. Numerical study of temperature dependent eigenfrequency responses of tilted functionally graded shallow shell structures // Struct. Eng. Mech. - 2018. -V. 68. - No. 5. - P. 527-536.
33. Madenci E. A refined functional and mixed formulation to static analyses of FGM beams // Struct. Eng. Mech. -2019. - V. 69. - No. 4. - P. 427-437.
34. Ahmed R.A., Fenjan R.M., Faleh N.M. Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections // Geomech. Eng. - 2019. -V. 17. - No. 2. - P. 175-180.
35. Vinyas M. On frequency response of porous functionally graded magneto-electro-elastic circular and annular plates with different electro-magnetic conditions using HSDT // Compos. Struct. - 2020. - V. 240. - P. 112044.
36. Hadji L. Influence of the distribution shape of porosity on the bending of FGM beam using a new higher order shear deformation model // Smart Struct. Syst. - 2020. - V. 26. -No. 2. - P. 253-262.
37. Sahoo B., Mehar K., Sahoo B., Sharma N., Panda S.K. Thermal frequency analysis of FG sandwich structure under variable temperature loading // Struct. Eng. Mech. - 2021. -V. 77. - No. 1. - P. 57-74.
38. Reddy J.N. A simple higher-order theory for laminated composite plates // J. Appl. Mech. - 1984. - V. 51. - No. 4. -P. 745-752. - https://doi.org/10.1115/L3167719
39. Touratier M. An efficient standard plate theory // Int. J. Eng. Sci. - 1991. - V. 29. - No. 8. - P. 901-916.
40. Soldatos K. A transverse shear deformation theory for homogeneous monoclinic plates // Acta Mech. - 1992. -V. 94. - No. 3. - P. 195-220.
41. Miller R.E., Shenoy V.B. Size-dependent elastic properties of nanosized structural elements // Nanotechnology. - 2000. -V. 11. - No. 3. - P. 139-147.
42. Jing G.Y., Duan H.L., Sun X.M., Zhang Z.S., Xu J., Li Y.D., Wang J.X., Yu D.P. Surface effects on elastic properties of silver nanowires: Contact atomic-force microscopy // Phys. Rev. B. - 2006. - V. 73. - No. 23.
43. Lu P., He L.H., Lee H.P., Lu C. Thin plate theory including surface effects // Int. J. Solid. Struct. - 2006. - V. 43. -No. 16. - P. 4631-4647.
44. Reddy J.N. Energy Principles and Variational Methods in Applied Mechanics. - John Wiley & Sons Inc., 2002.
45. Daouadji T.H., Hadji L. Analytical solution of nonlinear cylindrical bending for functionally graded plates // Geomech. Eng. - 2015. - V. 9. - No. 5. - P. 631-644.
46. Kiani Y. NURBS-based thermal buckling analysis of gra-phene platelet reinforced composite laminated skew plates // J. Thermal. Stress. - 2019. - P. 1-19.
47. Rachedi M.A., Benyoucef S., Bouhadra A., Bachir Bouiad-jra R., Sekkal M., Benachour A. Impact of the homogeniza-tion models on the thermoelastic response of FG plates on variable elastic foundation // Geomech. Eng. - 2020. -V. 22. - No. 1. - P. 65-80.
48. Merzoug M., Bourada M., Sekkal M., Ali Chaibdra A., Bel-mokhtar C., Benyoucef S., Benachour A. 2D and quasi 3D computational models for thermoelastic bending of FG beams on variable elastic foundation: Effect of the micro-mechanical models // Geomech. Eng. - 2020. - V. 22. -No. 4. - P. 361-374.
49. Hadji L., Zouatnia N., Bernard F. An analytical solution for bending and free vibration responses of functionally graded beams with porosities: Effect of the micromechanical models // Struct. Eng. Mech. - 2019. - V. 69. - No. 2. - P. 231241.
50. Safa A., Hadji L., Bourada M., Zouatnia N. Thermal vibration analysis of FGM beams using an efficient shear deformation beam theory // Earthquakes Struct. - 2019. - V. 17. -No. 3. - P. 329-336.
51. Zouatnia N., Hadji L. Effect of the micromechanical models on the bending of FGM beam using a new hyperbolic shear
deformation theory // Earthquakes Struct. - 2019. - V. 16. -No. 2. - P. 177-183.
52. Jena S.K., Chakraverty S., Malikan M., Mohammad-Sedi-ghi H. Hygro-magnetic vibration of the single-walled carbon nanotube with nonlinear temperature distribution based on a modified beam theory and nonlocal strain gradient model // Int. J. Appl. Mech. - 2020. - https://doi.org/10.1142/ s1758825120500544
53. Sedighi H.M., Bozorgmehri A. Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory // Acta Mech. - 2016. - V. 227. - No. 6. -P. 1575-1591.
54. Reddy J.N. Nonlocal theories for bending, buckling and vibration of beams // Int. J. Eng. Sci. - 2007. - V. 45. -P. 288-307.
55. Ould Youcef D., Kaci A., Houari M.S.A., Tounsi A., Benzair A., Heireche H. On the bending and stability of nanowire using various HSDTs // Adv. Nano Res. -2015. - V. 3. -No. 4. - P. 177-191.
56. AvcarM. Free vibration of imperfect sigmoid and power law functionally graded beams // Steel Compos. Struct. - 2019. -V. 30. - No. 6. - P. 603-615.
57. Sedighi H.M., Bozorgmehri A. Nonlinear vibration and adhesion instability of Casimir-induced nonlocal nanowires with the consideration of surface energy // J. Brazil. Soc. Mech. Sci. Eng. -2016. - V. 39. - No. 2. - P. 427-442.
58. Shariati A., Jung D. won, Mohammad-Sedighi H., Zur K.K., Habibi M., Safa M. On the vibrations and stability of moving viscoelastic axially functionally graded nanobeams // Materials. - 2020. - V. 13. - No. 7. - P. 1707.
59. AlSaid-Alwan H.H.S., Avcar M. Analytical solution of free vibration of FG beam utilizing different types of beam theories: A comparative study // Comput. Concret. - 2020. - V. 26. - No. 3. - P. 285-292.
60. Hadji L., Avcar M. Free vibration analysis of FG porous sandwich plates under various boundary conditions // J. Appl. Comput. Mech. - 2021. - V. 7. - No. 2. - P. 505-519.
Received 20.02.2021, revised 15.03.2021, accepted 18.03.2021
This is an excerpt of the article "Surface Effects and Small-Scale Impacts on the Bending and Buckling of Nanowires Using Various Nonlocal HSDTs". Full text of the paper is published in Physical Mesomechanics Journal. DOI: 10.1134/S1029959922010064
Сведения об авторах
Abdelmadjid Lounis, Post-Graduate, Université de Sidi Bel Abbés, Algeria, maimi2002fr@yahoo.fr Djamel Ould Youcef, Post-Graduate,Université de Sidi Bel Abbés, Algeria, d.ouldyoucef@gmail.com Abdelmoumen Anis Bousahla, Prof., Université de Sidi Bel Abbés, Algeria, bousahla.anis@gmail.com Fouad Bourada, Prof., University of Sidi Bel Abbes, Université de Tissemsilt, Algérie, bouradafouad@yahoo.fr Abdelhakim Kaci, Prof., University of Sidi Bel Abbes, Université Dr Tahar Moulay, Algérie, kaci_abdelhakim@yahoo.fr Houari Heireche, Prof., Université de Sidi Bel Abbés, Algeria, heireche_h@yahoo.fr
Abdeldjebbar Tounsi, Post-Graduate, University of Sidi Bel Abbes, Algeria, tounsi_abdeldjebbar@yahoo.com Kouider Halim Benrahou, Prof., University of Sidi Bel Abbes, Algeria, kbenrahouhalim@gmail.com
Abdelouahed Tounsi, Prof., University of Sidi Bel Abbes, Algeria, Yonsei University, Korea, King Fahd University of Petroleum and Minerals, Saudi Arabia, abdelouahed.tounsi@yonsei.ac.kr
Muzamal Hussain, Researcher, Government College University Faisalabad, Pakistan, muzamal45@gmail.com
