A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects

A non-classical model for the free vibrations of nanobeams accounting for surface stress effects is developed in this study. Based on Gurtin-Murdoch elasticity theory, the influence of surface stress is incorporated into the Euler-Bernoulli beam theory. A compact finite difference method (CFDM) of sixth order is employed for discretizing the non-classical governing differential equation to obtain the natural frequencies of nanobeams subject to different boundary conditions. To check the validity of the present numerical solution, based on an exact solution, an explicit formula for the fundamental frequency of simply-supported nanobeams is obtained. Good agreement between the results of exact and numerical solutions is achieved, confirming the validity and accuracy of the present numerical scheme. The comparison between the results generated by CFDM with those obtained by the conventional finite difference method (FDM) further reveals the advantages of the compact method over its classical counterpart. The influences of beam thickness, surface density, surface residual stress, surface elastic constants, and boundary conditions on the natural frequencies of nanobeams are also investigated. It is indicated that the effect of surface stress on the vibrational response of a nanobeam is dependent on its aspect ratio and thickness.

[1]  Reza Ansari,et al.  On the dynamic stability of embedded single-walled carbon nanotubes including thermal environment effects , 2012 .

[2]  Morton E. Gurtin,et al.  Surface stress in solids , 1978 .

[3]  Mehdi Dehghan,et al.  High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods , 2010, Math. Comput. Model..

[4]  R.K.N.D. Rajapakse,et al.  Analytical solutions for a surface-loaded isotropic elastic layer with surface energy effects , 2009 .

[5]  H. P. Lee,et al.  Application of nonlocal beam models for carbon nanotubes , 2007 .

[6]  A. I. Murdoch,et al.  The effect of interfacial stress on the propagation of stoneley waves , 1977 .

[7]  Hong-Bo Zhang,et al.  Higher-order compact finite difference method for systems of reaction-diffusion equations , 2009, J. Comput. Appl. Math..

[8]  Mohammad Taghi Ahmadian,et al.  A nonlinear Timoshenko beam formulation based on the modified couple stress theory , 2010 .

[9]  B. Tay,et al.  Transparent and flexible glucose biosensor via layer-by-layer assembly of multi-wall carbon nanotubes and glucose oxidase , 2007 .

[10]  Reza Ansari,et al.  Study of Small Scale Effects on the Nonlinear Vibration Response of Functionally Graded Timoshenko Microbeams Based on the Strain Gradient Theory , 2012 .

[11]  Tongqing Lu,et al.  The surface effect on the strain energy release rate of buckling delamination in thin film–substrate systems , 2011 .

[12]  Reza Ansari,et al.  BENDING BEHAVIOR AND BUCKLING OF NANOBEAMS INCLUDING SURFACE STRESS EFFECTS CORRESPONDING TO DIFFERENT BEAM THEORIES , 2011 .

[13]  Behrooz Farshi,et al.  Size-dependent longitudinal and transverse wave propagation in embedded nanotubes with consideration of surface effects , 2011 .

[14]  H. Hedayati,et al.  Influence of graded agglomerated CNTs on vibration of CNT-reinforced annular sectorial plates resting on Pasternak foundation , 2012, Appl. Math. Comput..

[15]  Marianna A. Shubov,et al.  Vibrational frequency distribution for nonconservative model of double-walled carbon nanotube , 2010, Appl. Math. Comput..

[16]  Reza Ansari,et al.  A sixth-order compact finite difference method for vibrational analysis of nanobeams embedded in an elastic medium based on nonlocal beam theory , 2011, Math. Comput. Model..

[17]  Leong Ming Gan,et al.  Synthesis of vertically aligned carbon nanotubes on metal deposited quartz plates , 2002 .

[18]  J. N. Reddy,et al.  A microstructure-dependent Timoshenko beam model based on a modified couple stress theory , 2008 .

[19]  H. P. Lee,et al.  Thin plate theory including surface effects , 2006 .

[20]  Xi-Qiao Feng,et al.  Effects of surface elasticity and residual surface tension on the natural frequency of microbeams , 2007 .

[21]  Ellad B. Tadmor,et al.  Surface effects in non-uniform nanobeams: Continuum vs. atomistic modeling , 2010 .

[22]  Reza Ansari,et al.  Nonlocal plate model for free vibrations of single-layered graphene sheets , 2010 .

[23]  S. Kitipornchai,et al.  Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory , 2009 .

[24]  Reza Ansari,et al.  Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory , 2011 .

[25]  Jun Luo,et al.  On the anti-plane shear of an elliptic nano inhomogeneity , 2009 .

[26]  M. Gurtin,et al.  Addenda to our paper A continuum theory of elastic material surfaces , 1975 .

[27]  M. A. Eltaher,et al.  Free vibration analysis of functionally graded size-dependent nanobeams , 2012, Appl. Math. Comput..

[28]  Yapu Zhao,et al.  The effects of surface tension on the elastic properties of nano structures , 2010 .

[29]  W. Spotz.  High-Order Compact Finite Difference Schemes for Computational Mechanics , 1995 .

[30]  Reza Ansari,et al.  Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain , 2010 .

[31]  Mehdi Dehghan,et al.  High-order compact boundary value method for the solution of unsteady convection-diffusion problems , 2008, Math. Comput. Simul..

[32]  Zhongmin Xiao,et al.  Analysis of a screw dislocation interacting with an elliptical nano inhomogeneity , 2009 .

[33]  Xiaogang Qu,et al.  A DNA nanomachine induced by single-walled carbon nanotubes on gold surface. , 2009, Biomaterials.

[34]  Feng Xu,et al.  Mechanical properties of ZnO nanowires under different loading modes , 2010 .

[35]  Yafei Zhang,et al.  Multi-wall carbon nanotube as a new infrared detected material , 2004 .

[36]  Abdelouahed Tounsi,et al.  Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field , 2010 .

[37]  Reza Ansari,et al.  Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models , 2012 .

[38]  Reza Ansari,et al.  A nonlinear Timoshenko beam formulation based on strain gradient theory , 2012 .

[39]  Lin Wang,et al.  Size-dependent vibration characteristics of fluid-conveying microtubes , 2010 .

[40]  Murat Sari,et al.  A sixth-order compact finite difference scheme to the numerical solutions of Burgers' equation , 2009, Appl. Math. Comput..

[41]  Yizhe Hu,et al.  Facile attachment of magnetic nanoparticles to carbon nanotubes via robust linkages and its fabrication of magnetic nanocomposites , 2009 .

[42]  Steven L. Crouch,et al.  Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes , 2009 .

[43]  K. M. Liew,et al.  Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures , 2007 .

[44]  Vijay B. Shenoy,et al.  Size-dependent elastic properties of nanosized structural elements , 2000 .

[45]  Morton E. Gurtin,et al.  The role of the configurational force balance in the nonequilibrium epitaxy of films , 2003 .