Bending Analysis of Carbon Nanotubes with Small Initial Curvature Embedded on an Elastic Medium Based on Nonlocal Elasticity and Galerkin Method

Document Type: Persian

Authors

1 MSc Student, Department of Mechanical Engineering, Isfahan University of Technology

2 Professor, Department of Mechanical Engineering, Isfahan University of Technology

Abstract

Carbon nanotubes have an important role in reinforcing nanocomposits. Many experimental observations have shown that in the most nanostructures such as nanocomposites, carbon nanotubes (CNTs) are often characterized by a certain degree of waviness along their axial direction. In the present paper, the effects of initial curvature, influence of surrounding medium that is modeled as Winkler elastic foundation on behavior of slightly curved carbon nanotubes are investigated. To capture the small size effects, nonlocal elasticity theory is implemented. Bending governing equations are derived using the principle of minimum total potential energy and these nonlinear equations are solved by Newton Raphson method. It is shown that the larger the initial curvature, the higher deflection can occur. Furthermore, neglecting the effect of initial curvature of CNTs can lead to incorrect results.

Keywords


[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.