آنالیز خمش نانولوله های دارای انحنای اولیه‌ی تعبیه شده بر روی بستر الاستیک بر مبنای تئوری الاستیسیته غیرمحلی و روش گالرکین

Subject Areas : Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering

اعظم عارفی ¹ , محمود سلیمی ²

1 - دانشجو، دانشکده مکانیک، دانشگاه صنعتی اصفهان
2 - استاد، دانشکده مکانیک، دانشگاه صنعتی اصفهان

Received: 2015-09-27 Accepted : 2015-09-27 Published : 2012-12-21

Keywords: تحلیل خمش, نانولوله ی کربنی با انحنای اولیه, تئوری الاستیسیته ی غیرمحلی, روش نیوتن رفسون, روش گالرکین,

Abstract :

نانولوله­های کربنی در تقویت کامپوزیت­ها نقش بسزایی ایفا می­کنند. بدیهی است که برخی از نانولوله­ها در هنگام کاربرد، شکل منظم ابتدایی خود را حفظ نمی کنند و دچار اعوجاج می­شوند. این اعوجاج می­تواند در حین فرآیند ساخت یا بعد از آن درنتیجه­ی تأثیر ماتریس رخ دهد. بر این اساس، مدل­سازی این نوع نانوساختار به صورت پوسته یا تیر بدون انحنا، می­تواند خطایی قابل ملاحظه را با نتایج همراه کند. در این مقاله، خمش نانولوله­های دارای انحنای اولیه مورد مطالعه قرار می­گیرد. معادلات تعادل بر پایه تئوری الاستیسیته غیر محلی به کمک اصل کمینه­سازی انرژی پتانسیل کل، استخراج و از روش گالرکین برای حل آنها بهره گرفته می­شود. از مدل وینکلر برای  مدل­سازی بستر الاستیک استفاده می­شود. حل خیز حاصل از دستگاه معادلات غیرخطی به کمک روش عددی نیوتن رفسون صورت  می­گیرد و در نهایت تأثیر مقیاس کوچک، انحنای اولیه و مدول فونداسیون بر روی خیز نانولوله­ها مورد بررسی قرار خواهد گرفت

References:

[1]  Iijima S., “Helical microtubules of graphitic carbon”, Nature, 354, 1991, pp. 56-58.

[2]  Thostenson E.T., Ren, Z., Chou, T.W., “Advances in the science and technology of carbon nanotubes and their composites: a review”, Composites Science and Technology, Vol. 61, 2001, pp. 1899–1912.

[3]  Zhou S.J., Li Z.Q., “Length scales in the static and dynamic torsion of a circular cylindrical micro-bar”, Shandong University Technology, Vol. 31, 2001, pp. 401–407.

[4]  Fleck N.A., Hutchinson J.W., “Strain gradient plasticity: theory and experiment”, Acta Metal Material, Vol. 42, 1994, pp. 475-487.

[5]   Yang A.C.M., Chong D.C.C., Lam P., “Couple stress based strain gradient theory for elasticity”, Solids Structure, Vol. 39, 2002, pp. 2731–2743.

[6]  Eringen A.C., “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves”, Journal of Applied Physics, Vol. 54, 1983, pp. 4703–4710,.

[7]  Lu P., Lee H.P., Lu C., Zhang P.Q., “Dynamic properties of flexural beams using a nonlocal elasticity model”, Journal of Applied Physics, Vol. 99, 2006, No. 073510.

[8]  Peddiseon P., Buchanan J.R., McNitt R.P., “Application of nonlocal continuum models to nanotechnology”, International Journal of Engineering Science, Vol. 41, 2003, pp. 305-312.

[9]  Lu P., Lee H.P., Lu C., Zhang P.Q., “Application of nonlocal beam models for carbon nanotubes”, International Journal of Solids Structure, Vol. 44, 2007, pp. 5289-5300.

[10] Reddy J.N., “Nonlocal theories for bending, buckling and vibration of beams”, International Journal of Engineering Science, Vol. 45, 2007, pp. 288-307.

[11]  Aydogdu M., “A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration”, Physica E, Vol. 41, 2009, pp. 1651-1655.

[12]  Lee H.L., Chang W.J., “Surface and small scale effects on vibration analysis of a nonuniform nanocantilever beam”, Physica E, Vol. 43, 2010, pp. 466-469.

[13]  Simsek M., “Forced vibration of an embedded single-walled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory”, Steel & Composite Structures, Vol. 11, 2011, pp. 59-76.

[14] Zhang Y.Q., Liu G.R., Han X., “Effect of Small Length Scale on Elastic Buckling of Multi-Walled Carbon Nanotubes Under Radial Pressure”, Physics Letters A, Vol. 69, 2006, pp. 370-376.

[15] Murmu T., Pradhan S.C., “Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM”, Physica E, Vol. 41, 2009, pp. 1232–1239.

[16]  khademolhosseini F., “Application of Nonlocal continuum shell models for torsion of single-walled carbon nanotubes”, Department of Mechanical Engineering, Sharif University of Technology, 2009.

[17] Senthilkumar V., “Buckling analysis of a single-walled carbon nanotube with nonlocal continuum elasticity by using differential transform method”, Advanced science letter, Vol. 3, 2010, pp. 337-340.

[18] Wang Yi-Ze., Li F.M., Kishimoto K., “Scale effects on thermal buckling properties of carbon nanotube”, Physics Letters A, Vol. 374, 2010, pp. 4890-4893.

[19]  Civalek O., Demir C., “Buckling and bending analysis of cantilever carbon nanotubes using the Euler-Bernoulli beam theory based on nonlocal continuum model”, Asian journal of Civil Engineering, Vol. 12, 2011, pp. 651-661.

[20] Wang C.M., Duan W.H., “Free vibration of nanorings/arches based on nonlocal elasticity”, Journal of Applied Physics, Vol. 104, 2008, 014303.

[21] Tepe A., “Nano-scale analysis of curved single walled carbon nanotubes for in-plane loading”, Journal of Computational and Theoretical Nanoscience, Vol. 7,2010, pp. 2405-2411.

[22] Duan W.H., Wang C.M., “Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on nonlocal plate theory”, Nanotechnology, Vol. 18, 2007, No. 385704.

[23]  Aghababaei R., Reddy J.N., “Nonlcal third order shear deformation plate theory with application to bending and vibration of plates”, Journal of Sound and Vibration, Vol. 326, 2009, pp. 227-289.

[24] Babaei H., Shahidi A.R., “Small scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method”, Archive Applied Mechanics, Vol. 81, 2010, pp.1051-1062.

 [25]  Ansari R., “Nonlocal finite element model for vibration of embedded multi layered graphene sheets”, Computational MaterialsScience, Vol. 49, 2010, pp. 831–838.

[26] Eringen A.C., “Nonlocal Continuum Field Theories”, Springer, NewYork, 2002.

[27]  Zhang Y.Q., Liu G.R., Xie X.Y., “Free transverse vibrations of double walled carbon nanotubes using a theory of nonlocal elasticity”, Physics Review B, Vol. 71(19), 2005, No. 195404.

[28] Wang Q., “Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum model”, Journal of Applied Physics, Vol. 98, 2005, No. 124301.

[29] Phadikar J.K., Pradhan S.C., “Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates”, Computational materials science, Vol. 49, 2010, pp. 492-499.

[30] Fung Y.C., Kaplan A., “Buckling of low arches or curved beams of small curvature”, NACA TN 2840,1952.

[31] Fisher F.T., Bradshaw R.D., Brinson, L.C., “Fiber waviness in nanotube-reinforced polymer composites-I: modulus prediction using effective nanotube properties”, Composite Science Technology, Vol. 63, 2003, pp. 1689-2391.

[32] Qian D., Dickey E.C., Andrews, R., Rantell, T., “Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites”, Applied physics Letter, Vol. 76(20), 2003, pp. 2868-2938.

[33] Bradshaw R.D., Fisher F.T., Brinson L.C., “Fiber waviness in nanotube-reinforced polymer composites—II: modeling via numerical approximation of the dilute strain concentration tensor”, Composite Science Technology, Vol. 63, 2003, pp. 1705-1727.

[34] Aydogdu M., “A general nonlocal beam theory: its application to nanobeam bending”, buckling and vibration,Physica E, Vol. 41, 2009, pp. 1651-1655.